Four-Dimensional Fano Toric Complete Intersections

FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS

T. COATES, A. KASPRZYK AND T. PRINCE

Abstract. We ﬁnd at least 527 new four-dimensional Fano manifolds, each of which is a complete intersection in a smooth toric Fano manifold.

1. Introduction Fano manifolds are the basic building blocks of algebraic geometry, both in the sense of the Minimal Model Program [7,14,23,50] and as the ultimate source of most explicit examples and constructions. There are finitely many deformation families of Fano manifolds in each dimension [36]. There is precisely one 1-dimensional Fano manifold: the line; there are 10 deformation families of 2-dimensional Fano manifolds: the del Pezzo surfaces; and there are 105 deformation families of 3-dimensional Fano manifolds [26,28,29, 40–44]. Very little is known about the classification of Fano manifolds in higher dimensions. In this paper we begin to explore the geography of Fano manifolds in dimension 4. Four-dimensional Fano manifolds of higher Fano index have been classified [16–19, 27, 30, 31, 35, 37, 52, 53]—there are 35 in total—but the most interesting case, where the Fano variety has index 1, is wide open. We use computer algebra to find many 4-dimensional Fano manifolds that arise as complete intersections in toric Fano manifolds in codimension at most 4. We find at least 738 examples, 717 of which have Fano index 1 and 527 of which are new. Suppose that Y is a toric Fano manifold and that L1,...,Lc are nef line bundles on Y such that −KY − Λ is ample, where Λ = c1(L1) + ··· + c1(Lc). Let X ⊂ Y be a smooth complete intersection defined by a regular section of ⊕iLi. The Adjunction Formula gives that −KX = −KY − Λ X so X is Fano. We find all four-dimensional Fano manifolds X of this form such that the codimension c is at most 4. Our interest in this problem is motivated by a program to classify Fano manifolds in higher dimensions using mirror symmetry [11]. For each 4-dimensional Fano manifold X as above, therefore, we compute the essential ingredients for this program: the quantum period and regularized quantum differential equation associated to X, and a Laurent polynomial f that corresponds to X under mirror symmetry; we also calculate basic geometric data about X, the ambient space Y , and f. The results of our computations in machine-readable form, together with full details of our implementation and all source code used, can be found in the ancillary files which accompany this paper on the arXiv.

2. Finding Four-Dimensional Fano Toric Complete Intersections Our method is as follows. Toric Fano manifolds Y are classiﬁed up to dimension 8 by Batyrev, Watanabe–Watanabe, Sato, Kreuzer–Nill, and Øbro. For each toric Fano manifold Y of dimension d = 4 + c, we:

arXiv:1409.5030v3 [math.AG] 16 Dec 2014 (i) compute the nef cone of Y ; 2 (ii) ﬁnd all Λ ∈ H (Y ; Z) such that Λ is nef and −KY − Λ is ample; (iii) decompose Λ as the sum of c nef line bundles L1,...,Lc in all possible ways. Each such decomposition determines a 4-dimensional Fano manifold X ⊂ Y , deﬁned as the zero locus of a regular section of the vector bundle ⊕iLi. To compute the nef cone in step (i), we consider dual exact sequences ρ 0 / L / ZN / Zd / 0

? D ρ 0 o L? o (ZN )? o (Zd)? o 0

Key words and phrases. Fano manifolds, mirror symmetry, quantum diﬀerential equations, Picard–Fuchs equations.

1 2 T. COATES, A. KASPRZYK AND T. PRINCE where the map ρ is defined by the N rays of a fan Σ for Y . There are canonical identifications L? =∼ H2(Y ; Z) =∼ Pic(Y ), and the nef cone of Y is the intersection of cones \ NC(Y ) = hDi : i 6∈ σi σ∈Σ N ? where Di is the image under D of the ith standard basis vector in (Z ) [15, Proposition 15.1.3]. The classes Λ in step (ii) are the lattice points in the polyhedron P = NC(Y ) ∩ −KY − NC(Y ) such that −KY − Λ lies in the interior of NC(Y ). Since NC(Y ) is a strictly convex cone, P is compact and the number of lattice points in P is finite. We implement step (iii) by first expressing Λ as a sum of Hilbert basis elements in NC(Y ) in all possible ways:

(1) Λ = b1 + ··· + br bi an element of the Hilbert basis for NC(Y ) where some of the bi may be repeated; this is a knapsack-style problem. We then, for each decomposition (1), partition the bi into c subsets S1,...,Sc in all possible ways, and define the line bundle Li to be the sum of the classes in Si. We found 117173 distinct triples (X; Y ; L1,...,Lc), with a total of 17934 distinct ambient toric varieties Y . Note that the representation of a given Fano manifold X as a toric complete intersection is far from unique: for example, if X is a complete intersection in Y given by a section of L1 ⊕ · · · ⊕ Lc then it is also 1 ? ? ? 1 a complete intersection in Y × P given by a section of π1 L1 ⊕ · · · ⊕ π1 Lc ⊕ π2 OP (1). Thus we have found far fewer than 117173 distinct four-dimensional Fano manifolds. We show below, by calculating quantum periods of the Fano manifolds X, that we find at least 738 non-isomorphic Fano manifolds. Since the quantum period is a very strong invariant—indeed no examples of distinct Fano manifolds X =6∼ X0 with the same quantum period GX = GX0 are known—we believe that we found precisely 738 non-isomorphic Fano manifolds. Eliminating the quantum periods found in [13], we see that at least 527 of our examples are new. Remark 2.1. There exist Fano manifolds which do not occur as complete intersections in toric Fano manifolds. But in low dimensions, most Fano manifolds arise this way: 8 of the 10 del Pezzo surfaces, and at least 78 of the 105 smooth 3-dimensional Fano manifolds, are complete intersections in toric Fano manifolds [12]. Remark 2.2. It may be the case that any d-dimensional Fano manifold which occurs as a toric complete intersection in fact occurs as a toric complete intersection in codimension d; we know of no counterex- amples. But even if this holds in dimension 4, our search will probably not find all 4-dimensional Fano manifolds which occur as toric complete intersections. This is because, if one of the line bundles Li involved is nef but not ample, then the Kählercone for X can be strictly bigger than the Kählercone for Y . In other words, it is possible for −KX to be ample on X even if −KY − Λ is not ample on Y . For an explicit example of this in dimension 3, see [12, §55].

3. Quantum Periods and Mirror Laurent Polynomials

The quantum period GX of a Fano manifold X is a generating function ∞ X d (2) GX (t) = 1 + cdt t ∈ C d=1 for certain genus-zero Gromov–Witten invariants cd of X which plays an important role in mirror symmetry. A precise definition can be found in [12, §B], but roughly speaking one can think of cd as the ‘virtual number’ of rational curves C in X that pass through a given point, satisfy certain constraints on their complex structure, and satisfy h−KX ,Ci = d. The quantum period is discussed in detail in [11, 12]; for us what will be important is that the regularized quantum period ∞ X d (3) GbX (t) = 1 + d!cdt t ∈ C, |t| ∞ d=1 satisfies a differential equation called the regularized quantum differential equation of X: m=N X m (4) LX GbX ≡ 0 LX = pm(t)D m=0 d where the pm are polynomials and D = t dt . It has been proposed that Fano manifolds should correspond under mirror symmetry to Laurent polynomials which are extremal or of low ramification [11], in the sense discussed in §4 below. An n-dimensional FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 3

±1 ±1 Fano manifold X is said to be mirror-dual to a Laurent polynomial f ∈ C[x1 , . . . , xn ] if the regularized quantum period of X coincides with the classical period of f: Z 1 1 dx1 dxn πf (t) = n ··· t ∈ C, |t| ∞ (2πi) (S1)n 1 − tf x1 xn If a Fano manifold X is mirror-dual to the Laurent polynomial f then the regularized quantum differential equation (4) for X coincides with the Picard–Fuchs differential equation satisfied by πf . The correspon- dence between Fano manifolds and Laurent polynomials is not one-to-one, but it is expected that any two Laurent polynomials f, g that are mirror-dual to the same Fano manifold are related by a birational × n × n transformation ϕ:(C ) 99K (C ) called a mutation or a symplectomorphism of cluster type [1, 20, 33]: ? ϕ ϕ f = g. We will write such a mutation as f 99K g. Mutations are known to preserve the classical period ϕ (ibid.): if f 99K g then πf = πg. Remark 3.1. In the paragraphs above we discuss the regularized quantum differential equation and the Picard–Fuchs differential equation. This involves choices of normalization. Our conventions are that the regularized quantum differential operator is the operator LX as in (4) such that:

(i) the order, N, of LX is minimal; and (ii) the degree of pN (t) is minimal; and (iii) the leading coeﬃcient of pN is positive; and (iv) the coeﬃcients of the polynomials p0, . . . , pN are integers with greatest common divisor equal to 1.

The Picard–Fuchs diﬀerential operator is the diﬀerential operator Lf such that: m=N X m Lf πf ≡ 0 Lf = Pm(t)D m=0 d where the Pm are polynomials and D = t dt , and that the analogs of conditions (i)–(iv) above hold.

We determined the quantum period GX , for each of the triples (X; Y ; L1,...,Lc) from §2, as follows. For each such triple we found, using the Mirror Theorem for toric complete intersections [22] and a generalization of a technique due to V. Przyjalkowski, a Laurent polynomial f that is mirror-dual to X. This process is described in detail in §5. We then computed, for each triple, the first 20 terms of the power series expansion of GbX = πf using the Taylor expansion: ∞ X d πf (t) = αdt d=0 d where αd is the coefficient of the unit monomial in f . We divided the 117173 triples into 738 “buckets”, according to the value of the first 20 terms of the power series expansion of GbX = πf , and then proved that any two Fano manifolds X, X0 in the same bucket have the same quantum period by exhibiting a chain ϕ0 ϕ1 ϕn−1 ϕn of mutations f 99K f1 99K ··· 99K fn 99K g that connects the Laurent polynomials f and g mirror-dual to X and X0. For each quantum period GX , we computed the quantum differential operator LX directly from the mirror Laurent polynomial f chosen above, using Lairez’s generalized Griffiths–Dwork algorithm [39]. PN m The output from Lairez’s algorithm is a differential operator L = m=0 Pm(t)D with P0,...,PN ∈ Q[t] such that, with very high probability, Lπf ≡ 0. Such an operator L gives a recurrence relation for the Taylor coefficients α0, α1, α2. . . of πf ; using this recurrence relation and the first 20 Taylor coefficients computed above, we solved for the first 2000 Taylor coefficients αk. We then consider an operator: N¯ X m L¯ = P¯m(t)D m=0 where the P¯m are polynomials of degree at most R¯, and impose the condition that Lπ¯ f ≡ 0. The 2000 Taylor coefficients of πf give 2000 linear equations for the coefficients of the polynomials P¯m and, provided that (N¯ + 1)(R¯ + 1) 2000, this linear system is highly over-determined. Since we are looking for the Picard–Fuchs differential operator (see Remark 3.1), we may assume that (N,¯ R¯) is lexicographically less than (N, deg pN ). We searched systematically for such differential operators with (N¯ + 1)(R¯ + 1) 2000, looking for the operator L¯ with lexicographically minimal (N,¯ R¯) and clearing denominators so that the analogs of conditions (iii) and (iv) in Remark 3.1 holds. We can say with high confidence that this operator L¯ is in fact the Picard–Fuchs operator Lf , although this is not proven—partly because Lairez’s algorithm relies on a randomized interpolation scheme that is not guaranteed to produce an operator annihilating 4 T. COATES, A. KASPRZYK AND T. PRINCE

πf , and partly because if Lf were to involve polynomials Pm of extremely large degree, 2000 terms of the Taylor expansion of πf will not be enough to detect Lf . The operators L¯ that we found satisfy a number of delicate conditions that act as consistency checks: for example they are of Fuchsian type (which is true for Lf , as Lf arises geometrically from a variation of Hodge structure). Thus we are confident ¯ 1 that L = Lf in every case . Since GbX = πf and LX = Lf by construction, this determines, with high confidence, the quantum period GX and the regularized quantum differential operator LX . Remark 3.2. The use of Laurent polynomials and Lairez’s algorithm is essential here. There is a closed formula [12, Corollary D.5] for the quantum period of the Fano manifolds that we consider, and one could in principle use this together with the linear algebra calculation described above to compute (a good candidate for) the regularized quantum differential operator LX . In practice, however, for many of the examples that we treat here, it is impossible to determine enough Taylor coefficients from the formula: the computations involved are well beyond the reach of current hardware, both in terms of memory consumption and runtime. By contrast, our approach using mirror symmetry and Lairez’s algorithm will run easily on a desktop PC. Remark 3.3. The regularized quantum differential equation for X coincides with the (unregularized) quantum differential equation for an anticanonical Calabi–Yau manifold Z ⊂ X. The study of the regularized quantum period from this point of view was pioneered by Batyrev–Ciocan-Fontanine–Kim– van Straten [5,6], and an extensive study of fourth-order Calabi–Yau differential operators was made in [2]. We found 26 quantum differential operators with N = 4; these coincide with or are equivalent to the fourth-order Calabi–Yau differential operators with AESZ IDs 1, 3, 4, 5, 6, 15, 16, 17, 18, 19, 20, 21, 22, 23, 34, 369, 370, and 424 in the Calabi–Yau Operators Database [57], together with one new fourth-order Calabi–Yau differential operator (which corresponds to our period sequence with ID 469).

4. Ramification Data Consider now one of our regularized quantum differential operators: m=N X m LX = pm(t)D m=0 as in (4), and its local system V → P1 \ S of solutions. Here S ⊂ P1 is the set of singular points of the regularized quantum differential equation. Definition 4.1 ([11]). Let S ⊂ P1 be a finite set and V → P1 \ S a local system. Fix a basepoint x ∈ P1 \ S. For s ∈ S, choose a small loop that winds once anticlockwise around s and connect it to x via a path, thereby making a loop γs about s based at x. Let Ts : Vx → Vx denote the monodromy of V along γs. The ramification of V is: X Ts rf(V) := dim Vx/Vx s∈S

The ramification rf(V) is independent of the choices of basepoint x and of small loops γs. A non-trivial, irreducible local system V → P1 \ S has rf V ≥ 2 rk V: see [11, §2]. Definition 4.2. Let V → P1 \ S be a local system as above. The ramification defect of V is the quantity rf(V) − 2 rk(V). A local system of ramification defect zero is called extremal.

Definition 4.3. The ramification (respectively ramification defect) of a differential operator LX is the ramification (respectively ramification defect) of the local system of solutions LX f ≡ 0.

To compute the ramification of LX , we proceed as in [13]. One can compute Jordan normal forms of the local log-monodromies {log Ts : s ∈ S} using linear algebra over a splitting field k for pN (t). (Every singular point of LX is defined over k.) This is classical, going back to Birkhoff [8], as corrected by Gantmacher [21, vol. 2, §10] and Turrittin [55]; a very convenient presentation can be found in the book of Kedlaya [34, §7.3]. In practice we use the symbolic implementation of Q provided by the computational algebra system Magma [9, 54]. We computed ramification data for 575 of the 738 regularized quantum differential operators, finding ramification defects as shown in Table1; this lends some support to the conjecture, due to Golyshev [11], that a Laurent polynomial f which is mirror-dual to a Fano manifold should have a Picard–Fuchs operator Lf that is extremal or of low ramification. For the remaining 163 regularized quantum differential operators, the symbol pN (t) contains a factor of extremely high degree. This makes the computation of ramification data prohibitively expensive.

1This could be proved in any given case using methods of van Hoeij [56]; cf. [39, §8.2.2]. FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 5

Ramiﬁcation defect 0 1 2 3 Number of occurrences 92 290 167 26

Table 1. Ramiﬁcation Defects for 575 of the 738 Regularized Quantum Diﬀerential Operators

5. The Przyjalkowski Method

We now explain, given complete intersection data (X; Y ; L1,...,Lc) as in §2, how to ﬁnd a Laurent polynomial f that is mirror-dual to X. This is a slight generalization of a technique that we learned from V. Przyjalkowski2 [47,48], and which is based on the mirror theorems for toric complete intersections due to Givental [22] and Hori–Vafa [24]. Recall the exact sequence

? D ρ 0 o L? o (ZN )? o (Zd)? o 0 ? N ? from §2 and the elements Di ∈ L , 1 ≤ i ≤ N, defined by the standard basis elements of (Z ) . Recall ? ∼ ? further that L = Pic(Y ), so that each line bundle Lm defines a class in L . Suppose that there exists a choice of disjoint subsets E, S1,. . . ,Sc of {1, 2,...,N} such that: ? •{ Dj : j ∈ E} is a basis for L ; • each Lm is a non-negative linear combination of {Dj : j ∈ E}; • P D = L for each m ∈ {1, 2, . . . , c}; k∈Sm k m ◦ and distinguished elements sm ∈ Sm, 1 ≤ m ≤ c. Set Sm = Sm \{sm}. Writing the map D in terms of N ? ? the standard basis for (Z ) and the basis {Dj : j ∈ E} for L defines an (N − d) × N matrix (mji) of × N integers. Let (x1, . . . , xN ) denote the standard co-ordinates on (C ) , let r = N −d, and define q1, . . . , qr and F1,...,Fc by: N Y mji X qj = xi Fm = xk i=1 k∈Sm Givental [22] and Hori–Vafa [24] have shown that: N Z V dxi tW i=1 xi (5) GX = e Vc Vr dqj Γ dFm ∧ m=1 j=1 qj × N where W = x1 + ··· + xN and Γ is a certain cycle in the submanifold of (C ) defined by:

q1 = ··· = qr = 1 F1 = ··· = Fc = 1 Sc ◦ Introducing new variables yi for i ∈ m=1 Sm, setting  1 P if i = sm  1+ ◦ yk x = k∈Sm i yi ◦ P if i ∈ S  1+ ◦ yk m k∈Sm and using the relations q1 = ··· = qr = 1 to eliminate the variables xj, j ∈ E, allows us to write W − c as a Laurent polynomial f in the variables: Sc ◦ Sc ◦ {yi : i ∈ m=1 Sm} and {xi : i 6∈ E and i 6∈ m=1 Sm}

The mirror theorem (5) then implies that GbX = πf , or in other words that f is mirror-dual to X. The Laurent polynomial f produced by Przyjalkowski’s method depends on our choices of E, S1,. . . ,Sc, and s1,. . . ,sc, but up to mutation this is not the case:

Theorem 5.1 ([46]). Let Y be a toric Fano manifold and let L1,...,Lc be nef line bundles on Y such that −KY − Λ is ample, where Λ = c1(L1) + ··· + c1(Lc). Let X ⊂ Y be a smooth complete intersection deﬁned by a regular section of ⊕iLi. Let f and g be Laurent polynomial mirrors to X obtained by applying Przyjalkowski’s method to (X; Y ; L1,...,Lc) as above, but with possibly-diﬀerent choices for the subsets ϕ E, S1,. . . ,Sc and the elements s1,. . . ,sc. Then there exists a mutation ϕ such that f 99K g.

2Przyjalkowski informs us that he learned this, for the case of the cubic threefold, from L. Katzarkov [32] and D. Orlov [49]. 6 T. COATES, A. KASPRZYK AND T. PRINCE

Example 5.2. Let Y be the projectivization of the vector bundle O⊕2 ⊕ O(1)⊕2 over P2. Choose a basis ? for the two-dimensional lattice L such that the matrix (mji) of the map D is: 1 1 1 0 0 1 1 0 0 0 1 1 1 1 ? Consider the line bundle L1 → Y deﬁned by the element (2, 1) ∈ L , and the Fano hypersurface X ⊂ Y deﬁned by a regular section of L1. Applying Przyjalkowski’s method to the triple (X; Y ; L1) with E = {3, 4}, S1 = {1, 2, 5}, and s1 = 1 yields the Laurent polynomial 2 (1 + y2 + y5) 1 + y2 + y5 f = + + x6 + x7 y2x6x7 y5x6x7 mirror-dual to X. Applying the method with E = {3, 4}, S1 = {1, 6}, and s1 = 1 yields: 2 (1 + y6) 1 + y6 g = x2 + + + x5 + x7 x2y6x7 x5y6x7 ϕ × 4 × 4 We have that f 99K g where the mutation ϕ:(C ) → (C ) is given by: x2 1 (x2, x5, y6, x7) 7→ , , x7, x2 + x5 = (y2, y5, x6, x7) x5y6 y6 Remark 5.3. Observe that, for a complete intersection of dimension n and codimension c, Przyjalkowski’s n+c method requires partitioning n + c variables into c disjoint subsets. If c < 2 then at least one of the subsets must have size one and so the corresponding variable, xj say, is eliminated from the Laurent polynomial via the equation xj = 1. One could therefore have obtained the resulting Laurent polynomial from a complete intersection with smaller codimension: new Laurent polynomials are found only when n+c c ≥ 2, that is, when the codimension is at most the dimension. In particular, all possible mirrors to 4-dimensional Fano toric complete intersections given by the Przyjalkowski method occur for complete intersections in toric manifolds of dimension at most 8.

6. Examples 6.1. The Cubic 4-fold. Let X be the cubic 4-fold. This arises in our classiﬁcation from the complete 5 5 intersection data (X; Y ; L) with Y = P and L = OP (3). The Przyjalkowski method yields [25, §2.1] a Laurent polynomial: (1 + x + y)3 f = + z + w xyzw mirror-dual to X, and elementary calculation gives: ∞ X (3d)!(3d)! π (t) = t3d f (d!)6 d=0

Thus GbX = πf , and the corresponding regularized quantum diﬀerential operator is: 4 3 2 2 LX = D − 729t (D + 1) (D + 2)

The local log-monodromies for the local system of solutions LX g ≡ 0 are: 0 1 0 0! 0 0 1 0 0 0 0 1 at t = 0 0 0 0 0 0 0 0 0! 0 0 0 0 1 0 0 0 1 at t = 9 0 0 0 0 0 0 0 0! 0 0 0 0 2 0 0 0 1 at the roots of 81t + 9t + 1 = 0 0 0 0 0 0 1 0 0! 0 0 0 0 0 0 0 1 at t = ∞ 0 0 0 0

and the operator LX is extremal. FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 7

6.2. A (3,3) Complete Intersection in P6. Let X be a complete intersection in Y = P6 of type (3, 3). 6 This arises in our classiﬁcation from the complete intersection data (X; Y ; L1,L2) with L1 = L2 = OP (3). The Przyjalkowski method yields a Laurent polynomial: (1 + x + y)3(1 + z + w)3 f = − 36 xyzw mirror-dual to X, and [12, Corollary D.5] gives: ∞ ∞ X X (3l)!(3l)!(k + l)! k k+l GbX = πf (t) = (−36) t k!(l!)7 k=0 l=0

The corresponding regularized quantum diﬀerential operator LX is: (36t + 1)4(693t − 1)D4 +18t(36t + 1)3(13860t + 61)D3 +9t(36t + 1)2(3492720t2 + 57672t + 77)D2 +144t(36t + 1)(11226600t3 + 377622t2 + 2754t + 1)D +15552t2(1796256t3 + 98496t2 + 1605t + 7)

The local log-monodromies for the local system of solutions LX g ≡ 0 are: 0 1 0 0! 0 0 1 0 0 0 0 1 at t = 0 0 0 0 0 0 0 0 0! 0 0 0 0 1 0 0 0 1 at t = 693 0 0 0 0 2 3 1 0 0! 2 0 3 0 0 1 1 at t = − 0 0 3 1 36 1 0 0 0 3

and so the operator LX is extremal.

7. Results and Analysis We close by indicating how basic numerical invariants—degree and size of cohomology—vary across the 4 738 families of Fano manifolds that we have found. The degree (−KX ) varies from 5 to 800, as shown in Figures1 and2. We do not have direct access to the size of the cohomology algebra of our Fano manifolds X, as many of the line bundles occurring in the complete intersection data (X; Y ; L1,...,Lc) are not ample and so the Lefschetz Theorem need not apply. But the order N of the regularized quantum diﬀerential operator is a good proxy for the size of the cohomology. N is the rank of a certain local system—an irreducible piece of the Fourier–Laplace transform of the restriction of the Dubrovin connection (in the Frobenius manifold • given by the quantum cohomology of X) to the line in H (X) spanned by −KX —and in the case where this local system is irreducible, which is typical, N will coincide with the dimension of H•(X). For our 4 examples, N lies in the set {4, 6, 8, 10, 12}. Figure3 shows how N varies with the degree (−KX ) , with darker grays indicating a larger number of examples with that N and degree. The isolated example on the right of Figure3, with N = 6 and degree 800, is the blow-up of P(1, 1, 1, 1, 3) 4 at a point. Figure4 again shows how N varies with the degree (−KX ) , but this time with toric Fano 4 manifolds highlighted in red. Figure5 shows how the Euler number χ(TX ) varies with the degree (−KX ) , with darker grays indicating a larger number of examples with that Euler number and degree. The three examples with the largest Euler number χ are a quintic hypersurface in P5, with χ = 825; a complete intersection of type (2, 4) in P6, with χ = 552; and a complete intersection of type (3, 3) in P6, with 1 3 3 χ = 369. The three examples with the most negative Euler number are P × V4 where V4 is a quartic 4 1 3 3 5 hypersurface in P , with χ = −112; P × V6 where V6 is a complete intersection of type (2, 3) in P , with 1 3 3 6 χ = −72; and P × V8 where V8 is a complete intersection of type (2, 2, 2) in P , with χ = −48. 8 T. COATES, A. KASPRZYK AND T. PRINCE

10 Frequency

0 0 100 200 300 400 500 600 700 800 Degree

Figure 1. The Distribution of Degrees (Frequency Plot)

700

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400 Frequency

300 Cumulative

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0 0 100 200 300 400 500 600 700 800 Degree

Figure 2. The Distribution of Degrees (Cumulative Frequency Plot)

12 10

N 8 6 4

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Figure 3. The Distribution of Degrees with N FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 9

12 10

N 8 6 4

200 400 600 800 Degree

Figure 4. The Distribution of Degrees with N, with Toric Fano Manifolds Highlighted.

800

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Euler number 300

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−100

200 400 600 800 Degree

Figure 5. The Distribution of Degrees with Euler Number. 10 T. COATES, A. KASPRZYK AND T. PRINCE

8. Source Code and Data The results of our computations, in machine readable form, together with full source code written in Magma [9] can be found in the ancillary files which accompany this paper on the arXiv. See the files called README.txt for details. The source code and data, but not the text of this paper, are released under a Creative Commons CC0 license [10]: see the files called COPYING.txt for details. If you make use of the source code or data in an academic or commercial context, you should acknowledge this by including a reference or citation to this paper.

Acknowledgments The computations underlying this work were performed using the Imperial College High Performance Computing Service and the compute cluster at the Department of Mathematics, Imperial College London. We thank Simon Burbidge, Matt Harvey, and Andy Thomas for valuable technical assistance. We thank John Cannon and the Computational Algebra Group at the University of Sydney for providing licenses for the computer algebra system Magma. This research was supported by a Royal Society University Re- search Fellowship (TC); the Leverhulme Trust; ERC Starting Investigator Grant number 240123; EPSRC grant EP/I008128/1; an EPSRC Small Equipment grant; and an EPSRC Prize Studentship (TP). We thank Alessio Corti for a number of very useful conversations, Pierre Lairez for explaining his generalized Griﬃths–Dwork algorithm and sharing his code with us, and Duco van Straten for his analysis of the regularized quantum diﬀerential operators with N = 4.

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Turrittin, Convergent solutions of ordinary linear homogeneous differential equations in the neighborhood of an irregular singular point, Acta Math. 93 (1955), 27–66. 56. Mark van Hoeij, Factorization of differential operators with rational functions coefficients, J. Symbolic Comput. 24 (1997), no. 5, 537–561. 57. Duco van Straten, Calabi-Yau Operators Database, online, access via http://www.mathematik.uni-mainz.de/CYequations/db/. 58. Keiichi Watanabe and Masayuki Watanabe, The classification of Fano 3-folds with torus embeddings, Tokyo J. Math. 5 (1982), no. 1, 37–48. 12 T. COATES, A. KASPRZYK AND T. PRINCE 7 0 0 0 0 0 α 2520 2520 3780 3780 5040 1260 Continued on next page. 6 0 0 0 90 90 90 90 90 90 α 900 900 as toric complete intersections, together with the are Taylor coefficients of the regularized quantum X ,... 1 , α 0 α 5 0 0 0 α 120 120 120 120 120 120 120 240 4 0 0 0 24 48 96 24 24 48 48 72 α . as a toric complete intersection. The regularized quantum period sequences, in X X considered in this paper, the description, degree, and first few terms of the regularized X 3 0 0 0 6 6 6 6 6 6 12 12 α 2 0 0 0 0 0 0 0 0 0 0 0 α 1 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 401 1 0 0 12 0 120 900 0 512 1 0594 0 1450 0 0 1459 0 0 1486 0 0 48 6 1405 0 0 6 1 0 0 6 0 0 0 6 0 24 6 0 48 120 0 48 120 0 90 120 90 90 0 90 1260 90 1260 1260 2520 2520 625 1 0 0 0 0 120 0 0 512 431 341 513 513 417 432 384 351 486 338 − . Ten terms of the Taylor expansion suffices to distinguish all the regularized quantum periods in2 : Table see the discussion d t d α , as well as a representative construction of X =0 2 ∞ d P 4 21 4 17 4 94 4 74 4 86 4 47 4 37 4 114 4 115 4 118 4 4 Regularized Quantum Period Sequences Arising From Complete Intersections in Toric Fano Manifolds P × P Q 2 BØS BØS BØS BØS P A. ) = t ( X b G Table 2: 738 regularized period sequences obtained from 4-dimensional Fano manifolds that arise as complete intersections in toric Fano manifolds. Appendix 3 5 7 9 BØS BØS 376 1 0 0 0 72 120 0 0 1 2 4 6 8 11 13 BØS 15 BØS 17 BØS 19 21 BØS 353 1335 0 1 0 0 0 6 6 48 72 120 120 90 90 5040 6300 10 12 14 16 18 20 22 Period ID Description ( In this Appendix we record, for each of the toric complete intersections period sequence: and supplementary table on page 37 .3 , 4 5 6 Tables and 7 contain constructions of our Fano manifolds description, degree, Euler number, and first two terms of the Hilbert series for lexicographic order, arepreviously shown known; in otherwise2 . Table the descriptions If are the exactly description as there in is [ 13 , left Appendix blank A]. then The no quantities Fano manifold with that regularized quantum period sequence was quantum period sequence for FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 13 7 α 840 840 840 3780 7560 1680 2520 3360 15120 15120 22680 15120 40320 22680 136080 Continued on next page. 6 20 20 α 900 540 900 540 380 380 380 380 1350 3240 3240 8100 20250 5 0 0 0 0 0 60 α 120 480 120 120 120 120 120 120 180 4 6 6 6 6 6 24 48 48 96 96 48 48 30 α 120 168 3 0 0 0 0 0 0 12 12 12 12 18 24 24 36 54 α 2 0 0 0 0 0 0 0 0 0 2 2 2 2 2 2 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 640 1560 0 1 2 0 2 0 0496 6 1 0 6 2 0 0 60 30 380 380 60 0 840 740 840 544 1 0 2 0 6 120 20 2520 512 1 0 2 0 30 0 740 0 322 1327 0 1 0 0 0 12 12 24 48 240 120 900 900 5040 7560 406364 1 0378 0230 297 12281 292 24244 242 0179 800 605 900624 489 446 3780 529 324 1 0243 0 1 0 24 0 0 36 0 0 0 3240 8100 0 0 249 1 0 0 18 72 360 2430 18900 − 3 4 2 4 1 P 4 12 4 10 4 32 4 30 4 46 4 87 4 18 4 31 4 15 4 121 4 105 4 4 4 3 × 1 BØS BØS BØS BØS BØS P 43 MW 45 BØS 53 BØS 25 27 BØS BØS 41 47 49 161BØS 51 1 0 0 72464 1 0 192 2 120 0 6 37800 180 211680 20 3360 23 24 26 BØS 28 30 32 34 36 38 40 42 44 46 48 50 52 29 31 33 FI BØS 35 37 FI 33039 1 0 0261 12 1 0211 0 72 1 0 24 0 120 96 36 144 540 120 120 3240 10080 8100 30240 75600 Continued from previous page. Period ID Description ( 14 T. COATES, A. KASPRZYK AND T. PRINCE 7 0 α 840 2520 3360 5040 2520 3360 2520 1680 1260 2940 3780 2940 3780 4200 20160 Continued on next page. 6 α 380 380 740 740 110 110 110 110 830 470 470 1100 1100 1100 1460 5060 5 0 60 60 60 α 120 180 240 120 180 120 960 120 180 240 120 180 4 6 6 6 6 30 30 30 54 54 54 54 78 30 30 30 α 150 3 0 0 0 0 0 0 0 0 0 6 6 6 6 6 6 6 α 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 431 1 0 2 0 30 120 740496 1 0 2520 2 6 30 60 470 2940 432 1 0 2 6 30 120 830 2520 415 1 0416 2 1 0 0 2 30 0 54560 180 1400 0 1442 0 2 0 1 740 2 0 6433 2 6 1 1100 0 6 6 3360 2 6 6 6 60 0 120 30 180 470 830 120 470 420 470 2520 2940 3780 464 433 376 480 370 400 352 345 240 576 592 480 432 432 463 432 − 3 2–33 4 8 4 7 4 6 4 11 4 26 4 20 4 24 4 41 4 15 4 45 4 82 4 12 4 13 4 104 4 106 4 109 4 111 MM BØS BØS BØS MW BØS BØS BØS × BØS BØS 1 P 63 430 1 0 2 0 54 120 740 1680 55 57 BØS 59 BØS 61 MW 65 67 34169 1 0 273 38375 0 1350BØS 077 1295 2BØS 079 30 1 2BØS 081 0BØS 283 0 360BØS 54 0 54 78 1820 120 240 480 1100 8400 1460 2540 2520 5040 10080 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 71 170 1 0 2 0 486 3600 18740 77280 Continued from previous page. Period ID Description ( FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 15 7 α 5460 6300 7560 4200 3780 5460 6720 6300 8820 1680 3360 5040 11340 14700 21000 Continued on next page. 6 α 830 470 830 830 920 560 560 1190 1190 1190 1190 1550 1910 2270 3350 5 60 α 180 240 240 120 180 180 240 360 360 420 660 180 240 360 4 6 6 6 30 30 30 54 54 54 54 54 54 78 α 102 126 3 6 6 6 6 6 6 6 6 6 6 6 6 12 12 12 α 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 432 1 0 2 12 6 240 560 2520 400 1411 0 1 2 0464 2 6 1 0 6 2 30368 30 6 1 0 180 2 54 240 6 470 60 54 470 830 180 5460 5040 830 2940 5880 384 369 337 416 390 384 357 357 324 303 272 254 432 417 352 − 3 1 4 70 4 52 4 13 4 92 4 16 4 71 4 91 4 81 4 27 F Q 4 113 × × 2 1 BØS BØS BØS BØS BØS BØS P P 87 89 BØS 91 93 BØS 95 97 37899 1BØS 0384 2 1353 0 6 1 2 0336 2 30 6 1 0 6 2 54 240 54 6 60 54 830 120 1190 180 1190 5040 1190 4200 5040 7980 85 86 88 BØS 90 92 94 96 98 101 103 336111 1336 0113 1 2 0115 2 6 6 54368 78 1320 0 1 240 2 0 180 2 12 12 1190 6 1190 6 7560 300 7980 420 920 920 4200 7980 105 107 109 308 1270 0 1480 2 0 1 2 0 6 2 6 12 78 102 6 420 600 120 1910 2990 920 10920 17640 840 100 102 104 106 108 110 112 114 Continued from previous page. Period ID Description ( 16 T. COATES, A. KASPRZYK AND T. PRINCE 7 α 4620 5460 5040 7560 8400 9240 1260 10080 11760 15540 14700 16800 17640 19320 22680 30240 Continued on next page. 6 α 920 1280 1280 1280 1280 1640 1280 2000 2000 2000 2720 2360 2000 2720 3800 1370 5 α 120 180 240 360 120 180 240 300 420 240 300 480 360 480 720 180 4 6 30 30 30 30 54 54 54 54 54 78 78 78 α 102 102 126 3 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 18 α 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 389 1 0347 2 1 0 12 2352 30 12 1 0 2 30 180 12 300 54 1280 1280 240 5460 1280 7980 9660 448 369 368 335 368 336 346 326310 1 0289 2304 273 12282 296 54266 249 240480 1640 10080 − 3 2–30 4 88 4 93 4 85 4 42 4 60 4 35 4 51 MM BØS BØS BØS × 1 P 117 119 BØS 121 123 BØS 125 127 352 1BØS 0305 2 1352 0 12 1 2 0 2 30 12 12 30 180 54 420 1640 120 1280 1640 5460 11760 8400 129 131 BØS 133 135 137 139 299141 1284 0143 1309 2 0145 1299 2 0 12 1288 2 0 12 1282 2 0 54 12 1212 2 0 54 12 1216 2 0 78 12 1 360 2 0 78 12 480 2 102 12 300 102 1640 12 360 102 2360 240 198 2000 480 2000 720 12600 2000 1200 16380 2720 14280 3800 15120 6320 18480 20160 35280 52920 116 118 120 122 124 126 128 130 132 134 136 138 140 142 144 146 Continued from previous page. Period ID Description ( FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 17 7 α 7140 1680 6720 8400 10080 13440 21000 56280 21840 35280 55440 105840 178920 287280 441840 Continued on next page. 6 α 2090 2090 2810 3170 5330 3260 3260 4340 4700 6140 9740 8120 15320 20360 43580 5 α 240 480 240 360 900 240 540 540 480 720 720 1440 1680 2640 1680 4 6 6 6 30 30 54 78 54 α 174 102 150 246 294 438 390 3 18 18 18 18 18 24 24 24 24 24 24 36 36 36 72 α 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 320 1 0 2 18 30 360 2090 7560 362 302 352304 1 0283 2218 320 18272 256 54260 230 180196 224 180 2090163 160 11340 − 3 2–28 4 73 MM × 1 P 149 159 161 304169 1282 0171 1 2 0173 2175 24177 24240 6 1202 0 54 1184 2 0 1192 360 2 0 36 1 360130 2 0 36 1 2 0 6 36 3260 2 150 3980 42 342 72 360 150 840 3360 870 1800 18480 900 8120 11000 5280 14240 14690 2520 62300 86520 188160 99540 1154160 151 153 BØS 155 157 304163 1256 0165 1200 2 0167 1 2 0 18 2 18256 78 18 1229 0 102 1213 2 0 246 1 300 2 0 24 360 2 1380 24 102 2450 24 3890 126 7850 174 420 660 18900 28560 960 85260 4700 5780 7220 35280 49560 70560 147 148 150 152 154 384156 1158 0160 2162 18164 166 6168 170 172 240174 176 1730 2100 Continued from previous page. Period ID Description ( 18 T. COATES, A. KASPRZYK AND T. PRINCE 7 0 α 3360 3360 3360 5460 7140 7140 5460 8400 8820 9660 16800 16800 20160 10500 2115960 Continued on next page. 6 α 400 400 490 490 490 850 1480 2560 2560 1930 1570 1210 1570 1930 1570 121880 5 0 60 60 α 360 360 120 180 240 240 480 180 240 240 240 300 6120 4 36 36 60 84 84 36 36 36 36 36 60 60 60 60 60 α 1062 3 0 0 0 0 0 6 6 6 6 6 6 6 6 6 6 α 108 2 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 432 1 0 4 6 36 240 490 7560 464 1 0384 4 1 0 0 4505 36 0 1382 0 1 4 0 60 120 4 6 6369 120 36 1 400 0 36367 4 1480 1 120 0 6 180 5040 4 60 6 850 5040 1210 60 180 2100 6720 240 1570 1570 6720 9660 448 1 0 4 0 60 0 1480 0 116 496 304 400 352 274 320558 1 0478 4447 409 0286 415400 108 1 0405 4368 0351 6352 60 3280 120 0 1570 4620 − 1 P 3 2–35 3 2–36 3 3–29 3 3–30 3 3–26 4 9 4 3 4 5 4 7 × 4 43 4 36 4 22 4 34 4 95 4 25 4 56 4 44 4 10 4 102 4 100 1 P MM MM MM MM MM BØS BØS MW BØS BØS BØS BØS × × × × × × 2 1 1 1 1 1 P P P P P P 181 BØS 185 187 BØS 191 193 BØS 195 320BØS 197 1 0 4201 203 0BØS 205 400207 84 1BØS 0 4353 240 6 1 0352 4 36 2560 1 0 6 4 300 60 6 10080 60 490 240 300 1210 9240 1210 10080 11760 179 183 98189 MW 1 0 2 180 1926199 BØS 11400 335360 6709920 178 180 182 184 186 188 190 192 194 196 198 200 202 204 206 208 Continued from previous page. Period ID Description ( FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 19 7 α 5880 8400 14700 10080 13440 17220 13860 15540 30660 11760 13440 21000 10080 10920 12180 Continued on next page. 6 α 940 1930 2290 2650 2650 3010 3370 3370 4810 1300 1300 1300 1660 2020 2380 5 α 360 120 240 300 360 240 300 660 360 420 480 600 300 360 360 4 60 84 84 84 84 36 36 36 36 60 60 60 α 108 108 132 3 6 6 6 6 6 6 6 6 12 12 12 12 12 12 12 α 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 378 1 0 4 12 36 360 1300 9660 416 1 0 4 12 36 360 940 8400 329 442368 1 0326 4320 310303 6 1 0274 4288 84256 6384 120368 84336 302 1930 360405 373352 1332 0321 2650 1 4620 4 0 4 12 12 15120 60 60 300 360 2020 2020 10080 13440 − 1 2 3 3–31 3 3–22 3 2–31 3 3–25 P F 4 48 4 29 4 58 4 66 4 54 × × MM MM MM MM 2 1 7 BØS × × × × S F 1 1 1 1 P P P P 213 368 1 0 4227 6 84 180 2290 6720 211 BØS 215 217 219 BØS 221 223 319225 1 0304 4229 1281 0231 6 1248 4 0233 1 4 0235 84 6 4237 6362BØS 239 84 6 1335 240BØS 0 108 1272 4 0 132 1 420 4 0 12 2650 240 4 12 600335 36 12 2650 1 0 36 4090 12180 4 36 4810 420 12 480 16800 14700 720 60 1300 29400 1300 940 360 11340 14700 2380 25200 13440 209 210 212 214 216 331218 1220 0222 4224 6226 228 60230 232 234 360236 238 1570 13860 Continued from previous page. Period ID Description ( 20 T. COATES, A. KASPRZYK AND T. PRINCE 7 α 14280 17220 18480 19320 15960 17640 18900 20580 20160 22260 27300 32760 29400 31080 37380 37800 Continued on next page. 6 α 2020 1660 2380 2020 3100 2380 3100 3460 3100 3460 3820 3100 3820 4180 4900 5260 5 α 420 480 480 540 360 420 420 420 480 480 600 720 540 600 720 720 4 60 60 60 60 84 84 84 84 84 84 84 84 α 108 108 108 132 3 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 α 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 336 321321 1 0314 4309 304 12331 325304 60 1 0302 4304 420293 12277 252 84 2020257 262 420256 16380 252 2740 17640 − 3 3–23 3 3–19 4 28 4 80 4 65 MM MM BØS × × 1 1 P P 241 243 BØS 245 247 249 251 320253 1289BØS 0255 1288 4 0257 1290 4 0259 12 1 4 0261 12289 4 60263 12 1304 0 60265 12 1289 4 0 60267 1 480272 4 0 84269 12 1 540266 4 0 12 1 600294 4 0 84 2020 12 1 360277 4 0 84 1660 12 1241 4 0 84 2020 12 1 420256 4 0 84 3100 12 17220 1 480 4 0 84 12 21840 480 4 108 3100 12 23520 600 108 2740 12 15960 660 108 3100 360 132 3460 540 19320 4180 720 19740 3820 600 21000 4180 28560 4180 31920 4900 18480 27720 38640 33600 240 242 244 246 248 250 252 254 256 258 260 262 264 266 268 270 Continued from previous page. Period ID Description ( FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 21 7 α 76860 10500 15540 22680 20580 25200 32340 30660 34020 39060 42840 55020 56280 77700 33600 Continued on next page. 6 α 9940 2110 2830 2830 3190 3910 4630 4270 4990 4990 5350 7150 7150 9310 3640 5 α 480 480 660 480 540 720 600 660 780 780 960 840 1200 1080 1080 4 36 60 60 84 84 84 36 α 180 108 108 108 132 132 156 180 3 12 18 18 18 18 18 18 18 18 18 18 18 18 18 24 α 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 304 1 0 4 18 60 600 2830 19740 194 394363 1 0341 4299 330 18310288 1299 0271 36 1 4 0283 4262 18 480257 18256 84235 84 1750225 480195 600240 10500 3550 3550 20580 25620 − 3 2–27 4 50 4 69 4 84 4 68 4 59 4 53 MM BØS BØS BØS × 1 P 273 275 BØS 277 279 281 283 331BØS 285 1BØS 0287 255 4289 1 0291 18 4293 257 36295 18 1277 0297 1257 4 0 60299 1 540256 4 0301 18 1240 4 0 18 1 840219 4 0 84 2110 18 1229 4 0 108 18 1288 4 0 108 3910 18 1 780256 4 0 108 18 600 13860 1 4 0 132 18 720 4 132 4270 24 780 32340 4630 156 24 840 4990 36 960 5350 84 1020 34020 5710 31920 720 7510 38220 720 7870 40320 48720 3640 57540 5800 63000 16800 31920 271 272 274 276 278 230280 1282 0284 4286 12288 290 132292 294 960296 298 300 7060 52920 Continued from previous page. Period ID Description ( 22 T. COATES, A. KASPRZYK AND T. PRINCE 7 0 α 7560 48720 53340 72240 54600 92400 58800 90720 13020 14280 18480 110880 223440 155400 231840 Continued on next page. 6 α 5800 6880 9040 8050 8500 1860 4740 1950 3390 3390 12280 11650 12820 20740 19390 33340 5 0 α 840 120 300 300 360 1080 1080 1440 1200 1320 1800 1200 2160 2040 2640 4 84 84 36 90 90 α 132 156 228 156 156 324 180 204 138 114 114 3 0 0 6 6 6 24 24 24 24 30 30 36 36 36 42 60 α 2 4 4 4 4 4 4 4 4 4 4 4 6 6 6 6 6 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 256 1 0 6 0 186 0 7980 0 384 1 0 6 0 114 0 3300 0 208 1 0 4 36 228368 1 0 1560 6 6 15340 114 122640 240 3390 9660 230 241 215 200 224 203 192 208 176 176 160 385 320 384 336 320 − 1 F 3 3–9 3 2–32 3 4–13 3 3–24 4 5 4 8 × 4 38 1 MM P MM MM MM MW × BØS × × × × 1 1 1 1 1 P P P P P 309 266319 1 0321 4323 325 30327 MW 193 84 1177 0 1144 4 0 1 840 4 0 42 4 42 156 6610 60 252 564 1680 2040 36540 4140 16510 21190 49900 119700 196980 648480 303 305 307 311 220313 1235 0315 1208 4 0317 1 4 0 24235 4 24 1208 0 108 24 1208 4 0 156 1 4 0 204 1080 30 4329 30 960 132331 1260 36 6520 228 7960 84 960 10480 1440 58800 1440306 8770 63420 1 12370 95760 0 6 10660 61740 6 116340 64680 114 300 3390 15540 302 304 306 308 310 312 314 316 318 320 322 324 326 328 330 332 Continued from previous page. Period ID Description ( FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 23 7 α 15540 24780 20160 21420 44520 22680 28560 32760 24360 33600 35280 36540 48720 44520 64680 Continued on next page. 6 α 4830 8070 2760 2760 4920 3840 3840 4200 5280 5280 5640 5640 5280 7080 8520 5 α 300 360 480 540 720 480 600 660 480 600 600 660 840 720 900 4 90 90 90 α 138 186 114 114 114 138 138 138 138 138 162 186 3 6 6 12 12 12 12 12 12 12 12 12 12 12 12 12 α 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 352 1320 0 1 6 0320 6 12 1 0 12 6 90288 90 12 1 0 540 6 114 600 12 2400 540 138 2400 3840 600 21420 26040 5280 23940 31080 304 250 364354 1 0336 6268 334 12304 288 90294 272 420286 272 242 2760256 210 17220 305 1 0 6 6 114 360 3750 18480 − 1 3 4–12 3 2–29 3 4–10 3 3–17 3 3–20 4 4 F 4 23 4 96 4 33 × MM MM MM MM MM 2 7 BØS BØS × × × × × S 1 1 1 1 1 P P P P P 335 337 339 BØS 341 343 288345 1 0347 6349 351 6258353 1 0355 138293 6357 1282 0359 12 1 360 6 0361 258 6 90363 12 1278 0 4830 12 1256 6 0 114 1 840228 6 0 114 12 1225 6 0 12 660 21000 1224 6 0 138 2400 12 720 1 6 0 138 12 6 3840 138 12 600 4200 162 12 660 41160 186 780 5640 186 32760 600 5280 36960 720 5640 900 7080 33600 8520 35280 8880 45360 38640 51240 63000 333 334 336 BØS 338 340 342 344 346 348 350 352 354 356 358 360 362 Continued from previous page. Period ID Description ( 24 T. COATES, A. KASPRZYK AND T. PRINCE 7 α 29820 38220 39480 39900 47460 52080 58380 65940 69720 76440 57120 46200 57960 78120 105840 206640 Continued on next page. 6 α 3210 4650 5730 6090 6090 5730 7530 8250 8970 9690 5820 6540 7980 9060 14280 23010 5 α 780 840 780 780 900 960 960 960 1320 1080 1080 1140 2340 1200 1080 1320 4 90 α 234 114 138 138 138 138 162 162 186 186 282 114 138 138 138 3 12 18 18 18 18 18 18 18 18 18 18 18 24 24 24 24 α 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 324 1 0 6 18 90 720 3570 28980 272 1 0 6 18 138 900 6090 46620 210 330 288 298 278 272 261 246 240 236 230 176 256 308298 1278 0246 1 6 0238 6 24 24 138 138 960 1080 6180 6540 46200 53760 − 2 3 3–16 3 3–18 3 2–25 P 4 49 4 63 4 57 4 55 × MM MM MM 2 6 BØS BØS × × × S 1 1 1 P P P 365 387 268 1 0 6 24 90 1080 5100 42840 367 369 371 373 375 293377 1267 0379 1288 6 0381 1 6 0383 18258 6385 18 1260 0 114 18 1226 6 0 114389 1239 6 0 138391 18 780 1226 6BØS 0393 18 960 1212 6BØS 0 138 18 780 1 6 5010 0 138 18 6 5010 162 18 900 5730 162 18 960245 186 34860 1020 1 6090 0 186 47040 1080 6 7170 39900 1140 8250 1560 24 8970 48720 8970 138 56700 13650 64260 71820 1260 74340 118860 7980 67620 364 366 368 370 372 374 376 378 380 382 384 386 388 390 392 394 Continued from previous page. Period ID Description ( FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 25 7 α 53760 80640 76440 78120 110880 107940 153720 199080 126420 159600 186480 131880 180600 137760 278040 Continued on next page. 6 α 7980 9420 9780 10140 12660 12660 16980 21300 14550 16710 19950 15360 16080 16080 27600 5 α 960 1320 1200 1260 1560 1500 1920 2280 1740 1980 2160 1920 2520 1800 2880 4 α 162 162 186 186 186 210 234 282 210 234 282 186 186 234 330 3 24 24 24 24 24 24 24 24 30 30 30 36 36 36 36 α 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 268 226 250 240 218 214 202 186 209 188 188 208 173 214 176 − 3 2–24 4 64 MM BØS × 1 P 405 407 409 411 413 208415 1209 0417 1178 6 0419 1214 6 0421 24 1204 6 0423 24 1194 6 0 210425 24 1244 6 0 210 30 1198 6 0 282 1440 30 1199 6 0 210 1620 30 1177 6 0 234 1920 36 1166 11940 6 0 282 1620 36 1 13020 6 0 186 1680 36 19140 6 186 1980 36 107520 13470 210 1560 36 115500 15270 306 2040 169260 18150 330 1800 116340 12480 2280 133560 15720 3240 168420 16440 23280 97440 138600 30480 136920 221760 312480 397 399 401 403 251 1230 0 1224 6 0 1230 6 0 24 1 6 0 24 6 162 24 162 24 186 1140 186 1320 1200 8700 1320 9780 10860 10500 67200 80640 82320 85680 395 396 398 400 402 220404 1406 0408 6410 24412 414 138416 418 1440420 422 424 7980 82320 Continued from previous page. Period ID Description ( 26 T. COATES, A. KASPRZYK AND T. PRINCE 7 0 α 19740 22260 39480 57960 59220 63000 72660 352800 270060 257040 406560 711480 441840 637560 117600 Continued on next page. 6 α 5120 5210 8810 6020 9620 6470 8270 34080 26970 28500 37860 60180 44520 57930 23300 10070 5 0 α 360 360 720 840 3480 2820 2760 3360 5880 4080 4860 1200 1080 1080 1140 4 α 378 306 282 426 570 354 474 168 168 216 168 216 360 168 192 216 3 0 6 6 36 42 48 48 48 60 66 12 12 12 18 18 18 α 2 6 6 6 6 6 6 6 8 8 8 8 8 8 8 8 8 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 307 1 0 8 0288 168 1 0 8 120 18 168 5120 1020 10080 6830 54600 304 1304 0 1288 8 0 1 8 0 6 8 12288 192 12 1 0 168 8 192 360 18 720 720 192 7010 5660 7460 1020 21000 39480 7910 42000 57120 166 172 176 160 140 151 145 384 336 282 294 256 186 282 272 256 − 1 P 1 × P 3 4–9 3 4–8 3 5–2 3 4–7 2 1 3 3–21 3 3–15 7 1 × F S P 4 39 1 × × × MM MM MM MM P MM MM 2 2 1 6 7 × × × × × × × P S S 1 1 1 1 2 7 1 1 × P P P P S P P 1 P 427 429 431 433 435 187437 1178 0439 1172 6 0441 1149 6 0 42 1162 6BØS 0 48 1160 6 0 306 48 1112 6 0 282 48 1 6 0 282451 2460 54 6 522 2760 60 378 120 3600 24090 474 4800 27420 1146 3480 32820 3960 229320 51180 11280 253680 38670 309120 45600 192300 595560 392700 503160 2817360 443 445 447 449 453 455 208 1 0 8256 12 1 0 8 288 18 1080 216 16100 1080 10070 92400 69300 426 428 430 432 434 436 438 440 442 444 446 448 450 452 454 456 Continued from previous page. Period ID Description ( FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 27 7 α 87360 78960 168000 110460 126840 154560 194880 142800 165900 173880 216720 239820 261240 295680 358680 Continued on next page. 6 α 8360 11150 11240 13400 15200 17360 22040 16370 17090 19340 21500 24380 27260 29780 35180 5 α 1260 1440 2160 1560 1680 1920 2160 1980 2160 2280 2700 2760 2760 3060 3360 4 α 216 168 216 240 264 264 312 264 264 264 288 312 360 360 408 3 18 24 24 24 24 24 24 30 30 36 36 36 36 36 36 α 2 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 240 1 0 8 24 240 1560 13040 105840 256 1 0 8 24 216 1440 10880 89040 240 252 192 230 220 208 192 224 204 209 197 188 184 182 162 − 3 4–5 3 3–13 3 3–11 MM MM MM × × × 1 1 1 P P P 463 465 467 469 471 473 219475 1214 0477 1204 8 0479 1230 8 0481 24 1224 8 0483 24 1192 8 0 240485 24 1182 8 0 264487 30 1203 8 0 288 1740 30 1188 8 0 216 1740 30 1182 8 0 264 1920 36 1176 13760 8 0 312 1800 36 1161 15920 8 0 264 1980 36 1 18440 8 0 312 2580 36 126420 13490 8 336 2880 36 133980 16730 360 2520 36 159600 23930 360 2760 116340 20420 432 2940 147000 22940 3300 229320 26180 3780 232680 28340 210420 31220 251160 39500 277200 320040 413280 459 461 224 1 0 8 18 240 1380 13310 105000 457 458 460 462 464 246466 1468 0470 8472 18474 476 216478 480 1200482 484 486 10430 79380 Continued from previous page. Period ID Description ( 28 T. COATES, A. KASPRZYK AND T. PRINCE 7 0 α 74760 91560 366240 410760 613200 608580 685440 887040 215040 186060 221760 134400 271740 273000 351960 Continued on next page. 6 α 27440 38960 53000 53810 60740 78740 15220 15760 12610 18460 21070 24670 15040 28000 28720 33400 5 0 α 960 4320 3960 4920 5160 5280 8040 1320 2400 2340 2520 2160 2880 3000 3480 4 α 264 408 504 480 552 408 318 318 270 318 342 366 270 366 390 414 3 0 48 48 48 54 60 84 12 18 24 30 30 36 36 36 36 α 2 8 8 8 8 8 8 10 10 10 10 10 10 10 10 10 10 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 288 1 0 10 12 270 840 11080 55440 230 1 0 10 0 270 240 10900 25200 192 1 0 8 48 360 3360208 31040 1 0 10 295680 30 342 2520 21430 208740 240 1 0 10 24 318 1800 17380 135240 160 167 141 151 148 129 256 240 252 188 224 204 270 206 198 172 − 1 P 3 4–4 3 2 4 2 3 2–18 3 2–19 3 2–23 3 3–12 7 5 × S 4 40 S B 1 × MM × P × MM MM MM MM 2 2 6 1 × × × × × × P S P 1 2 6 1 1 1 1 P S P P P P 489 491 493 495 497 499 156501 1161BØS 0 1151 8 0 1130 8 0507 48 1 8 0509 54 8 504511 60 360513 72 360515 4800180 792517 4200 1 0 5160176 51560 10 8460 1208 39770 0 24 1202 45260 10 0 1188 104120 572040 10 462 0 30 1 406980 10 0 36 514080 10 462 2640 36 1339800 366 36 366 2760 35740 414 2760 3000 35110 3180 287280 25840 26200 290640 31960 235200 260400 306600 503 505 218 1 0 10 12 366 960 20800 82320 488 490 492 494 496 498 500 502 504 506 508 510 512 514 516 518 Continued from previous page. Period ID Description ( FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 29 7 α 334320 436800 425040 465360 646800 711060 714000 945840 890400 186480 219240 1619940 1475880 1643880 1538880 Continued on next page. 6 α 33850 41770 38260 44740 57700 63910 59680 77320 74620 23160 24060 125290 113860 124030 124660 5 α 3480 4080 4320 4320 5280 5580 6120 7080 7200 9360 2160 2820 10080 10140 10560 4 α 414 462 414 486 558 582 510 654 846 558 846 846 702 396 396 3 42 42 48 48 48 54 60 60 66 72 72 78 96 24 36 α 2 10 10 10 10 10 10 10 10 10 10 10 10 10 12 12 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 224 1 0 12 24 444 2160 26760 191520 208 1 0 12 36 492 3360 35220 319200 188 172 172 172 156 146 150 144 133 142 130 125 128 216 240 − 3 4–3 2 2 3 3–10 6 5 S S × × MM MM 2 1 6 × × F S 1 1 P P 521 523 525 527 529 185531 1182 0533 1182 10 0535 1162 10 0 42 1167 10 0 48539 1146 10 414 0 48 1146 10 414 0 48 1140 10 462 0 3840 54 1 10 534 0 4080 54547 136 10 486 4200 60549 1 38530 606 0 4920 66 36460 10 654 4920 41140 750 6180 72 407820 53020 6840 387240 49150 726 7920 425880 68230 578760 77680 8280 534240 93970 805140 924840 97660 1156680 1212120 537 541 543 545 122 1 0130 10 1140 0 66 198 10 0 1 1038 10 78 0 10 84 13200 750 168 750 1566 8700 168850 8520 23040 107110 2289000 102880 402940 1328880 1244040 6002640 519 520 522 524 526 158528 1530 0532 10534 36536 538 510540 542 5280544 546 62560548 712320 Continued from previous page. Period ID Description ( 30 T. COATES, A. KASPRZYK AND T. PRINCE 7 α 371700 528360 577920 698460 898800 246540 468720 519120 594720 933240 1254960 1178520 1649340 2494800 1126440 Continued on next page. 6 α 38460 49710 54480 64020 77340 97320 33350 57200 59720 59540 82760 95450 101460 125490 155100 5 α 3540 4560 5040 6000 6840 8400 8340 2760 3960 4320 5280 7140 8160 10440 16320 4 α 492 540 588 636 684 780 780 876 756 546 690 714 690 786 834 3 36 42 48 60 60 60 72 78 30 36 36 48 60 66 α 108 2 12 12 12 12 12 12 12 12 12 14 14 14 14 14 14 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X K 192 1 0 12 48 564 4680 48000192 1 0 486360 14 54 690 5700 61070 631260 210 1 0 14 36 546 3480 37040 330540 176 1 0 12 60 636 6120 63300 693000 192 176 172 176 152 144 146 131 112 192240 1 0180 14176 192 0162 690156 0 50900 0 − 2 5 S 3 3–7 3 3–8 3 4–1 3 2 3 3 2–16 3 2–15 5 × S B 1 × MM MM MM × P MM MM 2 7 1 × × × × × × S P 1 1 1 1 1 1 P P P P P P 551 553 555 557 559 144 1 0168 12 1 0 36156 12569 1 540 0 48571 12573 588 5400 60575 577 708 5040 41700141 1154 0 6840 1 55200 14 0 705600 156 14 36 1 77700 0 36 588000 14 690 858 60 893760 5760 786 4560 59000 7320 83840 821520 84920 637560 981120 561 563 565 567 134 1136 0 1124 12 0 1 12 0 72 12 72579 708 90 876 1116 9120 9600146 13860 1 93000 0 118200 14 184350 72 1254960 1501920 882 2553600 9240 109940 1355760 550 552 554 556 558 560 562 564 566 568 570 572 574 576 578 Continued from previous page. Period ID Description ( FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 31 7 0 α 567840 1807680 1177260 1529640 2526720 3452400 4484760 4544400 9067800 1086540 1725780 2132760 2332680 3797640 4077780 Continued on next page. 6 α 98870 58390 97630 138020 122720 184820 237830 296420 311900 534200 160000 138490 162700 175750 261700 275470 5 0 α 8880 4920 8280 10800 10320 14040 17280 20640 21480 33000 11040 12600 13440 18600 19740 4 α 834 930 720 936 1002 1170 1338 1506 1506 2226 1296 1080 1152 1200 1488 1488 3 0 72 78 84 96 42 66 78 84 90 α 102 120 144 156 108 114 2 14 14 14 14 14 14 14 14 16 16 16 16 16 16 16 16 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X 98 K 160 1160 0 1 16 0144 16 72 1 0 84 1056 16 1104 108 9840 1248 11400 122920 15600 137860 188260 1428000 1685040 2538480 125 160 146 126 119 110 113 128 180 176 146 136 132 120 116 − 3 3–6 2 3 2–12 3 2–13 3 2–11 5 S 4 8 × MM MM MM MM V 2 6 × × × × S 1 1 1 1 P P P P 597 599 124 1114 0 1 16 0 16 24 60 1296 1344 4320 11520 163240 192940 840000 2347800 581 583 585 587 589 134591 1155 0593 1130 14 0595 1120 14 0 72 1113 14 0 78 1110 1026 14 0 84601 1103 14 906 0 96603 181 10560 1074 14 102 0605 1 1266 14 120 9600607 0 1530 12720 14 144 136220609 1554 15240 288 112190 19800 1506 163040140 20520 2994 1733760 207860 1 24480 284990 0 1363320 115 58440 16 2202060 298940 1 2918160 349700 0 90 4270980 16 1220900 4515000 1176 114 5456640 1488 21414960 12900 19440 164590 268990 2139060 3973620 580 582 584 586 588 590 592 594 596 598 600 602 604 606 608 610 Continued from previous page. Period ID Description ( 32 T. COATES, A. KASPRZYK AND T. PRINCE 7 0 α 3894240 1864800 3780420 6032880 7777560 8919960 1391880 5115600 8942640 5206320 9241680 18061680 73936800 14728980 Continued on next page. 6 α 268360 206820 262890 379980 478170 537120 969840 123050 342200 540740 347980 557140 824530 672000 3144440 5 0 α 9120 19440 19200 25200 30240 33240 55320 11760 23280 34080 23280 34200 47880 147120 4 α 1488 1494 1542 1878 2166 2310 2934 1188 1860 2340 7332 1914 2394 3090 3240 3 0 48 α 120 114 120 138 156 228 102 120 156 312 120 144 186 2 16 18 18 18 18 18 18 20 20 20 20 22 22 22 24 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X 99 84 72 95 96 K 150 1 0 20 60192 1 1140 0 22 9120 102 1434 121700 13740 1377600 160510 1881180 144 1 0 22 132 2058 24360 345280 4867800 120 120 125 104 104 216 120 102 128 112 − 3 3–3 2 1 2 5 P F S 4 6 × × × MM V 2 2 2 4 4 5 × S S S 1 P 625 633 639 641 83 193 0 1 22 0 24 246 36 4290 3240 74280 10800 1433830 680100 28650720 3528000 613 615 617 619 621 110623 1134 0 1112 16 0627 1110 18 126 0629 1100 18 102 0631 1 175290 18 120 0 1398 1 18 132635 0132 23940 1878 18 156637 1116 16200 1926 0 192 1100 23400 2190 20 355570 0 1 25800 2862 20 212670 120 0 32760 20 351180 126114 1668 5509980 46440 388800 168 1 1908 2919420 513720 0 21120 2580 5323080 22 802980 24480 6041280 162 38400 8536080 303320 2490 14515200 361010 629120 34260 4519200 5470920 10709160 531490 8504160 611 612 614 616 618 620 99 1622 0624 16626 114628 630 1512632 634 23640636 638 303190640 5285700 Continued from previous page. Period ID Description ( FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 33 7 α 7101360 2452380 9059820 3435600 6247500 14306040 32664240 25814460 74446680 31316460 44273880 21408240 79195200 80806320 60459840 44336040 Continued on next page. 6 α 720600 205530 578490 816000 270440 439590 1611960 1339550 3168440 1573390 2099350 1182900 3385200 3492320 2795400 2268560 5 α 21600 15720 37680 48120 83040 19080 72120 82200 27720 66960 151080 104100 153720 157800 134400 117840 4 α 3288 1704 2784 3192 4632 1998 4302 6222 4764 5580 2658 4338 7314 7728 7188 6954 3 72 α 102 174 192 264 108 246 396 258 306 126 240 372 384 360 348 2 24 24 24 24 24 26 26 26 28 28 30 30 30 32 36 38 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X 92 84 89 68 86 80 70 74 84 K 192 1 0 24 96 1704 14400 193920 2150400 128 1 0 30 216 3858 54000 891660 14726880 168 128 100 144 120 104 112 − 1 P 3 2–9 3 2–7 2 2 2 3 2–10 7 6 5 × S S S 1 × × × MM MM P MM 2 2 2 4 4 4 × × × × S S S 1 1 2 4 1 P P S P 649 651 655 657 94659 190661 0 1 24 074 26665 234108 1667 0 72 180 3648 26 0669 168 28 3534671 288 0 1 60780 28 240 070 5166 22320 28 288 3996 184 1060350 432 0 188 102960 5484 30 0 787580 62400 164 9660 32 300 0 1 19603500 100800 2038580 36 318 0 210240 6690 1067680 36 7514640 336 6144 2038960 552 44530080 124920 6708 5004640 19007520 113280 12852 42887040 119520 2778600 126134400 304080 2304770 2419200 61790400 7828200 48799800 50507520 210966000 643 645 647 653 83104 1 0 1 24 0 24 14482 192663 1 3480 0 3048 26 46920 216 45840 4302 909600 757680 72480 16450560 13077120 1371500 27676320 642 644 646 648 650 652 654 656 658 660 662 664 666 668 670 672 Continued from previous page. Period ID Description ( 34 T. COATES, A. KASPRZYK AND T. PRINCE 7 0 α 67735080 98491680 60897480 68151720 145945800 359059680 122340960 251895000 317604000 644873040 207233040 948659040 102194400 2223985680 Continued on next page. 6 α 3136160 5903540 5341780 7392000 9396300 4427820 3240110 7961600 3561250 4887190 12371480 11365000 20447820 28423860 58420920 5 0 α 150600 248880 445680 238560 367320 424440 207720 677520 171900 317100 181800 893280 225720 1614240 4 α 8010 11580 17388 11826 15120 15552 17412 11178 24378 10536 14424 11214 30156 13392 47574 3 0 α 396 516 696 528 660 696 528 888 498 600 498 522 1068 1356 2 38 44 44 46 48 48 52 54 54 56 56 58 60 64 66 α 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X 96 1 0 40 192 4776 59520 1120000 19138560 96 1 0 68 816 21012 465960 11662880 297392760 88 74 60 96 64 68 65 99 56 80 54 90 51 K 144 1 0 58 492 11214 178440 3502120 65938320 162 1 0 54 498108 9882 1 0 162000 60 504 2938770 11916 54057780 195120 3962040 78104880 144 126 − 1 P 3 2–5 3 2–6 2 2 2 2 1 2 4 6 7 5 × P F S S S S 4 1 4 × × × × × × MM MM P V 2 2 2 2 2 2 3 3 4 3 3 3 × × × S S S S S S 1 1 2 3 P P S 675 677 679 681 683 685 60687 154689 0 155 44691 0 161 44693 636 0 152 46695 888 0 1 15804 48697 714 064 23052 50699 216 393480 154 18618701 792 0 649200 196 15408 54 0 496560 10666340 1 21078 54 744 0 151320 19904120 56 1032 635760 1420381050 19194 301939680 528 37242 181 7959000 635293680 0 481680 18069260 1 11832 60 428469300 0 1222560 66 1212 117482400 217920 13279500 600739440 852 41404860 35916 4748600 21510 381906000 1134480 1477269360 504000 106293600 38512860 13009080 1368087000 347891040 703 673 674 676 678 680 682 80 1684 0686 38688 384690 692 8106694 696 156480698 700 3390500702 76130880 Continued from previous page. Period ID Description ( FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 35 7 α 368091360 185440080 909170640 804368880 1043489160 5023514160 1816414320 3245543280 17506424880 36647730000 28511659680 516454715280 824518956240 Continued on next page. 1685867765400 200205416839680 435882277192320 6 α 7862600 13640900 28577940 31974020 52448150 83881470 26609400 117404820 334814340 632940330 531501360 5996559080 9497959800 17014559940 1074550170260 1975803279600 5 α 520680 319560 937080 848880 1020360 2873160 1574580 2266320 6695280 11448480 10383840 73148160 114603120 180367920 5981882880 9412519800 4 α 22308 17550 34002 36240 76014 51492 67002 37260 146250 222918 219624 952176 1488708 2065764 35318526 48275736 3 α 852 588 984 1176 1212 2040 1626 1950 3408 4650 4752 13560 19992 24852 215808 262896 2 68 74 78 80 90 92 α 102 102 108 138 168 272 420 468 1946 2136 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 α 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 α 4 ) X 64 1 0 154 3840 159486 6504960 284808340 12889551360 80 1 0 92 1518 47172 135768048 1 0 42774050 398 17616 1385508600 1221810 85572960 6386359700 493612489440 78 72 63 60 45 62 53 38 54 35 40 26 30 26 32 17 K − 3 2–4 3 3 3 2 2 4 3 8 6 4 S S V V V × × MM × × × 2 2 3 3 1 1 1 × S S P P P 1 P 705 707 709 46 166 0 148 68719 0 1 78 1320721 0 78 1140723 43236 1680725 32706 1421040 6006642 877320 144 0 51100520 2142720 1 104 0 2620896036 128 2472 82424580 1914785040 1 2976 0 97944 814453920 184 120960 3324124440 5688 3940320 4959840 286008 171825080 221633120 14876160 7840793520 837897160 10369947840 49505030400 711 713 715 717 41 1 040 84 140 2148727 0 1 92729 0 77316 102 2112731 2688733 3051480 83820 106410 128188740 328128027 4495680 116 141863600 0 5649930720 120 203447460 444 0 1 1040 22404 0 6368328960 105984 1992 1771596 9658434240 227472 15564048 146305440 38459880 2472668160 6796332000 13047797460 422070022400 1282447706160 1221757064640 75673543680000 252711084477600 704 706 708 710 712 714 716 718 720 722 724 726 728 730 732 734 Continued from previous page. Period ID Description ( 36 T. COATES, A. KASPRZYK AND T. PRINCE 7 α 98894833331535360 2197113382189414080000 6 α 191430924575040 982577026659240000 5 α 389608626240 462392774925120 4 α 852143652 233995275000 3 α 2040912 130800000 2 α 6804 99000 1 0 0 α 0 1 1 α 4 ) X 9 5 K − 737 8 1 0 12816 5435904 3188239632 2051802731520 1419118168838400 1032164932439531520 735 736 738 12 1 0 2664 466368 115475112 31137505920 9021039724800 2746619333498880 Continued from previous page. Period ID Description ( FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 37

It appears from Table2 as if the regularized quantum period might coincide for the three pairs with period IDs 81 and 82, 117 and 118, and 248 and 249. This is not the case. The coeﬃcients α8, α9, and α10 in these cases are:

Period ID α8 α9 α10 81 10990 102480 365652 82 14350 87360 441252 117 32830 212520 1190952 118 32830 227640 1190952 248 120820 814800 5670504 249 127540 814800 6048504

Table 3: Certain 4-dimensional Fano manifolds with Fano index r = 1 that arise as complete intersections in toric Fano manifolds.

4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

738 5 825 6 21 1 1 1 1 1 1 5

737 8 552 7 27 1 1 1 1 1 1 1 2 4

736 9 369 7 28 1 1 1 1 1 1 1 3 3

735 12 324 8 34 11111111223

731 16 224 9 41 1111111112222

1 1 0 0 0 0 0 1 734 17 293 10 45 0 0 1 1 1 1 1 4

1 1 0 1 1 1 1 4 733 20 212 11 51 0 0 1 1 1 1 1 4

1 1 0 0 0 0 0 0 0 1 730 26 186 12 60 0 0 1 1 1 1 1 1 2 3

1 1 1 0 0 0 0 2 726 26 251 12 60 0 0 0 1 1 1 1 3

1 1 0 0 0 0 0 0 1 0 729 27 99 12 61 0 0 1 1 1 1 1 1 2 3

1 1 0 1 1 1 1 1 2 3 728 30 114 13 67 0 0 1 1 1 1 1 1 2 3

1 3 1 1 0 0 0 0 0 0 732 P × V4 32 −112 15 75 0 0 1 1 1 1 1 4

1 1 1 0 0 0 0 0 1 1 722 35 155 14 75 0 0 0 1 1 1 1 1 1 3

110000000001 725 36 92 14 76 001111111222

1 1 1 0 1 1 1 4 718 38 191 15 81 0 0 0 1 1 1 1 3

110111111222 724 40 72 15 83 001111111222

1 1 1 0 0 0 0 0 0 2 717 40 144 15 83 0 0 0 1 1 1 1 1 2 2

1 1 1 1 0 0 0 0 1 2 715 40 152 15 83 0 0 0 0 1 1 1 1 1 2

1 1 1 0 0 0 0 0 1 1 711 41 109 15 84 0 0 0 1 1 1 1 1 2 2

1 1 1 0 0 0 0 0 2 0 719 42 27 15 85 0 0 0 1 1 1 1 1 1 3

1 1 0 0 0 0 0 0 1 721 44 116 16 90 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 3

111000000110 712 45 63 16 91 000111111113

1 1 1 1 0 0 0 0 2 1 705 46 93 16 92 0 0 0 0 1 1 1 1 1 2

Continued on next page. 38 T. COATES, A. KASPRZYK AND T. PRINCE

Continued from previous page.

4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 3 1 1 0 0 0 0 0 0 0 0 727 P × V6 48 −72 18 100 0 0 1 1 1 1 1 1 2 3

1 1 1 0 1 1 1 1 2 3 709 48 99 17 97 0 0 0 1 1 1 1 1 1 3

111000000101 699 50 86 17 99 000111111122

1 1 1 0 0 1 1 3 702 51 135 18 103 0 0 0 1 1 1 1 3

1 1 1 1 0 1 1 4 685 52 168 18 104 0 0 0 0 1 1 1 2

1 1 0 0 0 1 1 1 3 716 53 80 18 105 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 3

2 2 1 1 1 1 0 0 0 0 3 0 720 S3 × S3 54 81 16 100 0 0 0 0 1 1 1 1 0 3

1 1 1 0 1 1 1 1 2 3 679 54 94 18 106 0 0 0 1 1 1 1 1 2 2

1 1 0 0 0 0 0 0 1 698 54 105 18 106 0 0 1 1 1 0 0 0 2 0 0 0 0 0 1 1 1 2

1 1 1 0 2 2 2 6 691 54 171 19 109 0 0 0 1 1 1 1 3

111100000111 681 55 85 18 107 000011111112

111000000200 690 56 40 18 108 000111111122

11000000010 710 60 12 19 115 00110000010 00001111113

11100000001100 678 60 48 19 115 00011111111122

111100000120 677 60 68 19 115 000011111112

1 0 0 0 0 0 1 1 683 61 149 20 119 0 1 1 1 1 1 1 4

1 1 0 0 0 0 0 1 1 714 62 44 20 120 0 0 1 1 1 1 0 0 3 0 0 0 0 0 0 1 1 1

1 1 0 0 1 0 1 1 3 708 63 48 20 121 0 0 1 1 0 1 1 1 3 0 0 0 0 1 1 1 1 3

1 3 110000000000 723 P × V8 64 −48 21 125 001111111222

111011111222 671 64 56 20 122 000111111122

11000000001 689 64 72 20 122 00110000001 00001111122

11000000010 686 65 37 20 123 00110000001 00001111122

1 1 0 0 0 1 1 1 3 707 66 84 21 127 0 0 1 1 0 1 1 1 3 0 0 0 0 1 1 1 1 3

1 1 0 0 0 0 1 1 2 684 68 62 21 129 0 0 1 1 1 0 0 0 2 0 0 0 0 0 1 1 1 2

1 1 1 0 0 1 1 1 2 2 661 68 64 21 129 0 0 0 1 1 1 1 1 2 2

1 1 1 1 0 1 1 1 2 3 656 68 93 21 129 0 0 0 0 1 1 1 1 1 2

11110000001110 666 70 52 21 131 00001111111112

Continued on next page. FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 39

Continued from previous page.

4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

11111000001111 665 70 60 21 131 00000111111111

2 2 111100000300 706 S3 × S4 72 72 20 130 000011111022

1 1 1 0 2 2 2 2 4 4 632 72 72 22 136 0 0 0 1 1 1 1 1 2 2

11000111122 676 74 42 22 138 00110000001 00001111122

11000000001 668 74 64 22 138 00111000011 00000111112

1 1 1 1 0 1 1 1 3 2 655 74 68 22 138 0 0 0 0 1 1 1 1 1 2

11000000101 704 78 0 23 145 00111110031 00000001101

1 1 0 0 0 0 0 0 0 1 3 713 P × MM2–4 80 −28 24 150 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 3

1100000000100 673 80 24 23 147 0011000000100 0000111111122

1 1 1 0 0 0 0 1 694 80 35 24 150 0 0 0 1 1 1 1 3

11000000010 660 80 40 23 147 00111000011 00000111112

11111000002110 659 80 40 23 147 00000111110112

1 1 0 0 1 1 1 1 3 701 81 36 24 151 0 0 1 0 0 1 1 1 3 0 0 0 1 1 1 1 1 3

1 1 1 1 0 0 1 3 595 81 101 24 151 0 0 0 0 1 1 1 2

1 0 0 0 0 0 1 1 0 2 653 82 84 24 152 0 1 1 1 1 1 1 1 2 3

1 1 1 0 0 0 1 1 3 639 83 72 24 153 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 1 2

1 0 0 0 0 0 1 1 1 1 645 83 77 24 153 0 1 1 1 1 1 1 1 2 3

11001011122 670 84 32 24 154 00110111122 00001111122

111101111222 624 84 52 24 154 000011111112

11100000002 650 84 54 24 154 00011100011 00000011111

1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 667 84 58 24 154 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 2

11000000001 658 86 21 24 156 00111000020 00000111112

11000000101 674 88 32 25 161 00111110022 00000001101

11000111122 669 88 40 25 161 00110111122 00001111122

11000011112 654 89 40 25 162 00111000011 00000111112

Continued on next page. 40 T. COATES, A. KASPRZYK AND T. PRINCE

Continued from previous page.

4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 0 0 0 0 0 1 1 2 0 651 90 12 25 163 0 1 1 1 1 1 1 1 2 3

11100000011 623 90 45 25 163 00011100011 00000011111

11000000010 2 2 700 S3 × S5 90 63 24 160 00111000020 00000111103

1 0 0 0 0 0 0 1 0 1 642 92 78 26 168 0 1 1 1 1 1 1 1 2 3

1 0 0 0 0 0 0 1 1 0 641 93 51 26 169 0 1 1 1 1 1 1 1 2 3

11011000011 649 94 32 26 170 00111000011 00000111112

1100000000001 638 95 29 26 171 0011100000110 0000011111112

1 1 0 0 0 0 0 0 0 1 3 680 P × MM2–6 96 −24 27 175 0 0 1 1 1 0 0 0 2 0 0 0 0 0 1 1 1 2

1 1 0 0 0 1 1 1 3 1 3 703 P × MM2–5 96 −12 27 175 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 3

1 1 1 0 0 0 0 0 1 0 693 96 −9 27 175 0 0 0 1 1 1 1 1 1 3

2 2 11111000002200 675 S4 × S4 96 64 25 169 00000111110022

1 1 1 1 0 0 1 1 2 2 545 98 52 27 177 0 0 0 0 1 1 1 1 1 2

1 1 1 0 0 0 0 1 2 594 98 55 27 177 0 0 0 1 1 1 0 0 2 0 0 0 0 0 0 1 1 1

1 1 1 1 0 2 2 5 179 98 125 28 180 0 0 0 0 1 1 1 2

1 1 1 0 1 1 1 3 688 99 27 28 181 0 0 0 1 1 1 1 3

1 1 0 0 0 1 0 1 2 622 99 40 27 178 0 0 1 1 1 0 1 1 3 0 0 0 0 0 1 1 1 2

111110111222 611 99 51 27 178 000001111111

11000011112 648 100 26 27 179 00111000020 00000111112

1100000000010 631 100 34 27 179 0011100000110 0000011111112

1100000000001 621 100 36 27 179 0011110000111 0000001111111

11100000020 630 102 21 27 181 00011100011 00000011102

1 1 0 0 0 0 0 0 1 593 103 63 28 185 0 0 1 0 0 0 0 1 1 0 0 0 1 1 1 1 1 3

1100000001001 664 104 12 28 186 0011111100221 0000000011001

11100011122 620 104 37 28 186 00011000001 00000111112

Continued on next page. FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 41

Continued from previous page.

4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 0 0 0 0 0 1 1 2 0 0 1 1 0 0 0 0 0 1 647 104 40 28 186 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 2

111110111222 618 104 40 28 186 000001111012

11001111122 657 108 20 29 193 00100111122 00011111122

10000100000 01111000003 2 2 697 S3 × S6 108 54 28 190 00000101010 00000011000 00000000110

1100001111112 619 110 26 29 195 0011100000110 0000011111112

1110000000110 613 110 26 29 195 0001110000101 0000001111012

11100011122 591 110 34 29 195 00011000010 00000111112

1110000000101 590 110 34 29 195 0001110000011 0000001111111

11000000001 1 3 672 P × MM2–7 112 −8 30 200 00110000000 00001111122

1 1 1 0 0 0 0 0 0 1 636 112 24 30 200 0 0 0 1 1 1 1 1 2 2

1 1 1 0 0 0 0 2 439 112 52 30 200 0 0 0 1 1 1 1 2

1 1 0 0 0 0 1 1 2 617 112 56 30 200 0 0 1 1 1 0 1 1 3 0 0 0 0 0 1 1 1 2

1 1 0 0 0 0 0 1 566 112 68 30 200 0 0 1 1 1 1 1 3

1 1 0 1 1 0 1 1 3 592 113 40 30 201 0 0 1 1 1 0 1 1 3 0 0 0 0 0 1 1 1 2

1 1 1 0 0 0 0 1 2 589 113 45 30 201 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 1 2

1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 637 114 31 30 202 0 0 0 0 1 1 1 0 0 2 0 0 0 0 0 0 0 1 1 1

100000011011 599 114 42 30 202 011111111222

1100000111111 609 115 27 30 203 0011110000111 0000001111111

110000000010 001100000010 629 116 20 30 204 000011000001 000000111112

1110000000200 610 116 22 30 204 0001110000011 0000001111111

1 1 1 1 0 2 2 2 3 4 178 116 68 31 207 0 0 0 0 1 1 1 1 1 2

11100001112 588 119 34 31 210 00011100011 00000011102

1101100000110 612 120 24 31 211 0011100000110 0000011111112

Continued on next page. 42 T. COATES, A. KASPRZYK AND T. PRINCE

Continued from previous page.

4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

100000011002 614 120 24 31 211 011111111222

11000101112 587 120 26 31 211 00111011122 00000111112

110000000001 001100000001 608 120 36 31 211 000011100011 000000011111

11000000001 00110000001 628 120 40 31 211 00001100001 00000011001 00000000111

1100000000100 2 2 662 S4 × S5 120 56 30 208 0011100000200 0000011111022

1 1 1 0 0 0 2 2 4 537 122 53 32 216 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 1 2

11100000111 565 124 27 32 218 00011110012 00000001101

100000001001 597 124 36 32 218 011111111222

1 1 0 0 0 0 1 0 1 2 0 0 1 1 0 0 0 1 1 2 616 125 28 32 219 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 2

1 1 0 0 0 0 1 0 1 580 125 29 32 219 0 0 1 1 1 1 0 1 3 0 0 0 0 0 0 1 1 1

11100001112 542 125 34 32 219 00011100011 00000011111

110000000010 001100000001 586 126 27 32 220 000011100011 000000011111

1 0 0 0 0 0 1 0 0 0 2 2 0 1 1 1 1 0 0 0 0 3 696 S3 × S7 126 45 32 220 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 1 0

11000000000 1 3 646 P × MM2–9 128 −8 33 225 00111000011 00000111112

11000111122 1 3 663 P × MM2–10 128 0 33 225 00110000000 00001111122

111000000100 634 128 4 33 225 000111111122

1 1 1 1 1 0 0 1 2 2 544 128 40 33 225 0 0 0 0 0 1 1 1 0 2

1 1 1 1 1 0 0 1 2 2 498 129 41 33 226 0 0 0 0 0 1 1 1 1 1

11100001121 585 130 23 33 227 00011100002 00000011111

1 1 1 0 0 1 0 0 2 540 130 26 33 227 0 0 0 1 1 0 1 1 2 0 0 0 0 0 1 1 1 2

11100000111 497 130 26 33 227 00011110021 00000001101

11000000001 541 130 34 33 227 00111101122 00000011111

1 1 1 1 0 1 2 4 177 130 68 34 230 0 0 0 0 1 1 1 2

Continued on next page. FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 43

Continued from previous page.

4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

111000000001100 564 131 22 33 228 000111000000011 000000111111111

1 1 1 0 1 1 1 1 2 2 627 132 12 34 232 0 0 0 1 1 1 1 1 2 2

110000000010 001100000010 606 132 22 33 229 000011100002 000000011111

1 1 1 0 1 1 0 1 3 536 133 35 34 233 0 0 0 1 1 1 0 0 2 0 0 0 0 0 0 1 1 1

11000011112 581 134 34 34 234 00111011122 00000111112

1 1 0 0 0 0 0 1 1 2 0 0 1 1 0 0 0 1 1 2 615 134 36 34 234 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 2

1 1 0 0 1 1 1 1 3 561 134 38 34 234 0 0 1 0 0 0 0 1 1 0 0 0 1 1 1 1 1 3

111000000001100 563 136 20 34 236 000111000001010 000000111110112

111000000001100 539 136 22 34 236 000111100001011 000000011110111

110000011112 001100000010 604 136 24 34 236 000011000010 000000111112

11011011122 543 140 24 35 243 00111011122 00000111112

1 1 1 0 0 2 0 2 4 434 140 26 35 243 0 0 0 1 1 0 1 1 2 0 0 0 0 0 1 1 1 2

1 1 0 0 0 1 1 0 0 2 0 0 1 1 0 0 0 0 1 1 605 140 26 35 243 0 0 0 0 1 1 1 0 0 2 0 0 0 0 0 0 0 1 1 1

1 1 1 0 0 0 0 0 1 2 0 0 0 1 1 0 0 0 0 1 535 140 36 35 243 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1

1 0 1 1 1 1 0 0 3 571 141 −3 35 244 0 1 1 1 1 1 0 1 3 0 0 0 0 0 0 1 1 0

1 1 1 0 0 0 1 0 2 492 141 33 35 244 0 0 0 1 1 1 0 1 2 0 0 0 0 0 0 1 1 1

11000000010 538 142 20 35 245 00111101122 00000011102

1 1 1 0 0 0 1 1 3 1 3 607 P × MM2–11 144 −8 36 250 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 2

1 1 0 0 0 0 0 0 0 1 1 3 0 0 1 1 0 0 0 0 0 1 635 P × MM3–3 144 4 36 250 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 2

1 1 1 1 0 0 0 0 1 2 323 144 23 36 250 0 0 0 0 1 1 1 1 1 1

1 1 0 0 0 0 0 0 1 560 144 24 36 250 0 0 1 1 1 0 0 0 2 0 0 0 0 0 1 1 1 1

1 1 1 0 0 0 0 1 2 534 144 34 36 250 0 0 0 1 1 0 1 1 2 0 0 0 0 0 1 1 1 2

Continued on next page. 44 T. COATES, A. KASPRZYK AND T. PRINCE

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 0 0 0 0 0 1 0 2 692 S3 × F1 144 36 36 250 0 0 1 1 1 1 0 0 3 0 0 0 0 0 0 1 1 0

1 1 0 1 1 1 1 3 551 144 36 36 250 0 0 1 1 1 1 1 3

1 1 0 0 0 0 0 0 0 2 1 1 695 S3 × P × P 144 36 36 250 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 3

1000001000000 0111110000022 2 2 652 S4 × S6 144 48 35 247 0000001010100 0000000110000 0000000001100

10000000101 438 145 33 36 251 01110000011 00001111112

110000000101 001100000010 602 146 16 36 252 000011110021 000000001101

1110000111112 533 146 20 36 252 0001110000110 0000001111012

1110000111211 531 146 22 36 252 0001110000011 0000001111111

1 1 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 0 0 2 584 146 23 36 252 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 1 1 0 2 0 0 1 1 0 0 1 0 1 2 579 146 24 36 252 0 0 0 0 1 1 0 1 1 2 0 0 0 0 0 0 1 1 1 2

11000010111 530 146 24 36 252 00111101122 00000011111

110000001111 001100000001 562 146 25 36 252 000011100011 000000011111

11100001112 496 148 22 36 254 00011100020 00000011102

1 0 0 0 1 0 1 1 2 433 149 34 37 258 0 1 1 1 1 0 1 1 3 0 0 0 0 0 1 1 1 2

11000000001 532 150 32 37 259 00100000101 00011111122

110000000001 2 2 001100000010 625 S5 × S5 150 49 36 256 000011100020 000000011102

1110110000111 494 151 22 37 260 0001110000011 0000001111111

11000000010 495 151 25 37 260 00100000101 00011111122

10000000110 436 151 25 37 260 01110000011 00001111112

11000000000100 00110000000100 558 152 22 37 261 00001110000011 00000001111111

1 1 0 0 0 0 0 1 1 573 154 30 38 266 0 0 1 1 1 1 0 1 3 0 0 0 0 0 0 1 1 1

Continued on next page. FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 45

Continued from previous page.

4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 0 0 0 0 1 1 1 2 0 0 1 1 0 0 0 0 0 1 583 155 24 38 267 0 0 0 0 1 0 0 1 1 2 0 0 0 0 0 1 1 1 1 2

11100010112 528 156 20 38 268 00011101112 00000011102

1110000001101 491 156 20 38 268 0001111100121 0000000011001

1 1 0 0 0 0 1 0 1 2 0 0 1 1 0 0 1 0 1 2 577 156 24 38 268 0 0 0 0 1 1 0 1 1 2 0 0 0 0 0 0 1 1 1 2

110000000101 001110000011 559 156 24 38 268 000001110011 000000001101

11000000011 00110000001 578 156 28 38 268 00001100001 00000011001 00000000111

1 1 1 0 0 0 0 2 2 519 158 25 39 273 0 0 0 1 1 1 0 0 2 0 0 0 0 0 0 1 1 1

1100000000000 1 3 601 P × MM2–12 160 0 39 275 0011110000111 0000001111111

1100000000000 1 3 603 P × MM2–13 160 4 39 275 0011100000110 0000011111112

1 0 0 1 1 0 0 1 2 437 160 24 39 275 0 1 1 1 1 0 0 1 2 0 0 0 0 0 1 1 1 2

1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 1 1 582 160 24 39 275 0 0 0 0 1 1 1 0 0 2 0 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 0 0 0 1 488 160 30 39 275 0 0 1 1 1 1 1 1 2 2

1 1 1 0 0 0 0 0 1 1 322 160 30 39 275 0 0 0 1 1 1 1 1 1 2

1 1 0 0 0 0 0 0 1 432 160 32 39 275 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 2

1 1 1 1 1 0 2 2 2 4 176 160 72 40 278 0 0 0 0 0 1 1 1 0 2

11100000111 487 161 22 39 276 00011100011 00000011111

11001111122 493 161 24 39 276 00100001111 00011111122

1 0 0 0 0 0 1 1 41 161 69 40 279 0 1 1 1 1 1 2 4

110000000101 001110000020 576 162 19 39 277 000001110011 000000001101

10000000101 435 162 19 39 277 01110000020 00001111112

11100010121 486 162 21 39 277 00011101112 00000011111

11000000000100 00111000000110 527 162 22 39 277 00000111000101 00000000111011

Continued on next page. 46 T. COATES, A. KASPRZYK AND T. PRINCE

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

2 2 1 1 1 0 0 0 0 0 687 S3 × P 162 27 40 280 0 0 0 1 1 1 1 3

1 1 1 1 0 1 1 3 174 163 31 40 281 0 0 0 0 1 1 1 2

11101110122 426 166 20 40 284 00011110021 00000001101

11101101122 425 166 22 40 284 00011100011 00000011111

110000010111 001100001111 529 167 22 40 285 000011100011 000000011111

1 1 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 1 490 167 24 40 285 0 0 0 0 1 1 1 0 1 2 0 0 0 0 0 0 0 1 1 1

1100000000010 0011000000010 555 168 24 40 286 0000110000001 0000001100001 0000000011111

100000010000 2 2 011111000022 644 S4 × S7 168 40 40 286 000000110100 000000001100

1 1 1 1 0 3 3 6 71 170 116 43 297 0 0 0 0 1 1 1 2

110001110021 001100000101 554 172 20 41 293 000011110021 000000001101

11000001001 518 172 20 41 293 00111110122 00000001101

1111000011121 428 172 20 41 293 0000111000101 0000000111011

110000001111 001101100011 526 172 22 41 293 000011100011 000000011111

111000001121 000110000001 522 172 22 41 293 000001100001 000000011111

11001111122 524 172 24 41 293 00100001102 00011111122

1 1 1 0 0 0 2 2 4 431 172 58 42 296 0 0 0 1 1 0 1 1 2 0 0 0 0 0 1 1 1 2

1 0 1 1 1 0 0 0 2 420 173 23 41 294 0 1 1 1 1 0 1 1 3 0 0 0 0 0 1 1 1 1

1 1 0 0 0 0 0 0 0 1 3 572 P × MM2–15 176 −4 42 300 0 0 1 0 0 0 0 1 1 0 0 0 1 1 1 1 1 3

11100011122 1 3 557 P × MM2–16 176 4 42 300 00011000000 00000111112

11000000001 552 176 9 42 300 00111000010 00000111112

1 1 0 0 0 0 0 1 1 2 1 3 0 0 1 1 0 0 0 0 0 0 600 P × MM3–6 176 12 42 300 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 2

Continued on next page. FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 47

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 1 0 0 0 0 0 2 0 430 176 16 42 300 0 0 0 1 1 1 1 1 1 2

111100000110 318 176 16 42 300 000011111112

1 1 0 0 0 0 0 1 1 1 0 0 1 1 1 0 0 0 0 2 556 176 20 42 300 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1

1 1 0 0 0 0 1 1 2 485 176 20 42 300 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 1 2

10001011112 424 176 20 42 300 01111011122 00000111112

11100002222 386 176 22 42 300 00011100011 00000011111

11000001111 511 176 24 42 300 00111101122 00000011111

1 1 1 0 0 1 1 1 3 320 176 33 42 300 0 0 0 1 0 0 0 1 1 0 0 0 0 1 1 1 1 2

11001111012 423 177 21 42 301 00110000111 00001111112

1000010000001 321 177 21 42 301 0111110000111 0000001111111

11100010011 409 178 20 42 302 00011101112 00000011111

1111000011112 429 178 22 42 302 0000111000110 0000000111002

1 1 1 1 1 0 2 2 3 3 40 179 41 43 306 0 0 0 0 0 1 1 1 1 1

1 1 1 1 0 1 0 1 3 172 180 32 43 307 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 1

11100001112 507 180 36 43 307 00011101112 00000011102

1 1 0 0 0 0 0 1 1 2 0 0 1 1 0 0 0 1 1 2 570 180 40 43 307 0 0 0 0 1 1 0 1 1 2 0 0 0 0 0 0 1 1 1 2

100000100000 011000000001 2 2 000111000002 598 S6 × S5 180 42 42 304 000000101010 000000011000 000000000110

1 1 0 0 0 1 1 0 1 2 0 0 1 1 0 0 1 1 1 2 525 182 20 43 309 0 0 0 0 1 0 0 1 1 2 0 0 0 0 0 1 1 1 1 2

110000001111 001100001111 523 182 22 43 309 000011100011 000000011111

11001111122 477 182 22 43 309 00100000101 00011111122

111000000111 000110000001 484 182 22 43 309 000001110011 000000001101

Continued on next page. 48 T. COATES, A. KASPRZYK AND T. PRINCE

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 2 483 182 24 43 309 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1

111000001112 000110000010 482 184 24 43 311 000001100010 000000011102

1 0 0 1 1 0 2 2 4 173 184 40 44 314 0 1 1 1 1 0 2 2 4 0 0 0 0 0 1 1 1 2

1 1 1 0 0 0 1 1 2 521 185 17 44 315 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 1 2

1 1 1 0 0 1 0 1 2 410 186 20 44 316 0 0 0 1 0 0 1 1 2 0 0 0 0 1 1 1 1 2

1 1 1 0 0 0 0 1 2 450 186 33 44 316 0 0 0 1 1 1 0 1 2 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 1 1 1 2 0 0 1 1 0 0 0 0 0 1 427 187 24 44 317 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 2

10111110022 506 188 8 44 318 01111110122 00000001100

11100001011 414 188 19 44 318 00011110112 00000001101

11000000001001 00111000000110 517 188 19 44 318 00000111100111 00000000011001

11000000011011 00111000000110 481 188 20 44 318 00000111000101 00000000111011

110000011011 001100010111 480 188 21 44 318 000011101112 000000011111

111000000111 000110000010 416 188 22 44 318 000001110011 000000001101

11000000101 00110000001 520 188 24 44 318 00001100001 00000011011 00000000111

1 1 1 0 0 0 0 1 2 1 3 489 P × MM2–18 192 4 45 325 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 2

110000000001 1 3 001100000000 553 P × MM3–7 192 12 45 325 000011100011 000000011111

11100000011 550 192 12 45 325 00011100020 00000011101

1 1 0 0 0 0 0 0 1 1 1 3 0 0 1 1 0 0 0 0 0 0 575 P × MM3–8 192 16 45 325 0 0 0 0 1 1 1 0 0 2 0 0 0 0 0 0 0 1 1 1

11000000001 00110000001 1 3 574 P × MM4–1 192 16 45 325 00001100001 00000011000 00000000111

Continued on next page. FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 49

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

11100000011 470 192 18 45 325 00011100001 00000011111

111100000111 175 192 18 45 325 000011111111

1 1 0 1 1 1 1 1 2 2 462 192 20 45 325 0 0 1 1 1 1 1 1 2 2

1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 1 475 192 22 45 325 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1

1 1 1 0 1 1 1 3 314 192 30 45 325 0 0 0 1 1 1 1 2

11000000100 2 633 S4 × F1 192 32 45 325 00111110022 00000001100

11000000000 2 1 1 643 S4 × P × P 192 32 45 325 00110000000 00001111122

1000000001001 319 193 19 45 326 0111000000110 0000111111112

1 1 1 0 0 0 2 0 2 272 194 19 45 327 0 0 0 1 1 1 0 1 2 0 0 0 0 0 0 1 1 1

1 0 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 1 1 415 194 19 45 327 0 0 0 0 1 1 1 0 0 2 0 0 0 0 0 0 0 1 1 0

1110011100111 298 195 18 45 328 0001100011111 0000011111111

1 1 1 1 0 1 1 1 2 2 168 196 16 46 332 0 0 0 0 1 1 1 1 1 2

1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 0 0 1 2 478 197 24 46 333 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1

1000000001010 419 198 18 46 334 0111000000110 0000111111112

11000000011 00110001001 516 198 24 46 334 00001100001 00000011001 00000000111

1 0 0 0 1 1 1 0 0 2 0 1 1 0 0 0 0 0 0 1 421 199 15 46 335 0 0 0 1 1 1 1 0 1 2 0 0 0 0 0 0 0 1 1 0

11101101122 308 200 24 46 336 00011100020 00000011102

1 1 1 1 0 0 0 2 3 157 200 32 47 339 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 1

1 1 1 0 0 0 1 0 1 2 0 0 0 1 1 0 0 0 0 1 408 202 22 47 341 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1

1 0 0 0 0 1 0 1 1 171 202 24 47 341 0 1 1 1 1 1 0 1 3 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 0 1 1 2 0 0 1 0 0 1 0 1 1 2 515 202 28 47 341 0 0 0 1 1 1 0 1 1 2 0 0 0 0 0 0 1 1 1 2

Continued on next page. 50 T. COATES, A. KASPRZYK AND T. PRINCE

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

110000011111 001110000011 479 203 19 47 342 000001001111 000000111111

11101101122 312 203 20 47 342 00011101112 00000011111

1110000000001100 0001110000001010 469 204 18 47 343 0000001110000101 0000000001110011

1 1 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 1 0 1 474 204 20 47 343 0 0 0 0 1 1 1 0 1 2 0 0 0 0 0 0 0 1 1 1

110000000010 001100000010 413 204 20 47 343 000010000101 000001111112

1100000000101 0011000000010 510 204 22 47 343 0000110000010 0000001110011 0000000001101

1 1 1 0 0 0 0 1 1 514 206 13 48 348 0 0 0 1 1 1 0 0 2 0 0 0 0 0 0 1 1 1

1 1 1 0 0 0 2 2 4 1 3 317 P × MM3–9 208 4 48 350 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 2

11000000000 1 3 509 P × MM2–19 208 4 48 350 00111101122 00000011111

11000011112 468 208 14 48 350 00111000010 00000111112

1 1 0 0 0 0 1 0 1 2 1 3 0 0 1 1 0 0 0 1 1 2 549 P × MM3–10 208 16 48 350 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 2

1 1 0 0 0 0 1 1 1 313 208 16 48 350 0 0 1 1 1 0 0 0 2 0 0 0 0 0 1 1 1 1

111000000110 315 208 16 48 350 000111111112

11000000001 307 208 18 48 350 00111000011 00000111111

1 1 0 0 0 1 1 1 2 316 208 20 48 350 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 2

1 1 0 0 0 0 0 1 1 405 208 20 48 350 0 0 1 1 1 1 0 0 2 0 0 0 0 0 0 1 1 1

11100001121 449 208 20 48 350 00011101112 00000011111

11101101122 418 208 22 48 350 00011101112 00000011102

11000000011 00110000011 513 208 24 48 350 00001100001 00000011001 00000000111

110000001001 001110000011 412 209 19 48 351 000001110111 000000001101

Continued on next page. FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 51

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

111000000111 000110000010 407 209 20 48 351 000001100001 000000011111

1100000010111 0011000001111 476 209 23 48 351 0000110000010 0000001100001 0000000011111

1 0 1 1 0 0 1 1 2 362 210 20 48 352 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 2

10000001000 01100000001 2 2 569 S7 × S5 210 35 48 352 00011100002 00000011010 00000000110

1 1 1 0 1 0 2 2 4 364 210 36 49 355 0 0 0 1 1 0 1 1 2 0 0 0 0 0 1 1 1 2

1 0 0 0 0 0 1 1 39 211 29 49 356 0 1 1 1 1 1 1 3

11100222022 143 212 16 48 354 00011000111 00000111111

1 1 1 0 0 0 0 0 2 2 0 0 0 1 1 0 0 0 0 1 385 212 22 49 357 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1

11110110122 167 213 19 49 358 00001110011 00000001101

110000000101 001110110021 467 214 19 49 359 000001110011 000000001101

1 1 0 0 0 1 1 0 1 2 0 0 1 1 0 0 0 1 0 1 411 214 20 49 359 0 0 0 0 1 1 1 0 1 2 0 0 0 0 0 0 0 1 1 1

110000110011 001110000111 406 214 20 49 359 000001110011 000000001101

1 1 0 0 0 1 0 1 1 2 0 0 1 1 0 0 1 1 1 2 422 214 24 49 359 0 0 0 0 1 0 0 1 1 2 0 0 0 0 0 1 1 1 1 2

1 0 0 0 1 1 0 0 0 1 0 1 1 0 0 0 0 0 0 1 306 215 21 49 360 0 0 0 1 1 1 0 1 1 2 0 0 0 0 0 0 1 1 1 1

1 1 1 1 0 0 1 2 145 216 16 50 364 0 0 0 0 1 1 1 2

2 2 1 1 1 0 0 0 0 0 0 0 626 S4 × P 216 24 50 364 0 0 0 1 1 1 1 1 2 2

100001000000 011000000000 001010000001 2 2 000110000000 546 S6 × S6 216 36 49 361 000001010100 000000110000 000000001100 000000000011

1 1 0 0 0 1 0 1 2 404 218 14 50 366 0 0 1 1 1 0 1 1 2 0 0 0 0 0 1 1 1 2

1 1 1 1 0 0 2 0 3 156 218 20 50 366 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 1 1 1

Continued on next page. 52 T. COATES, A. KASPRZYK AND T. PRINCE

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

11000000101 503 218 20 50 366 00111110122 00000001101

1 1 1 0 1 1 0 0 2 295 219 19 50 367 0 0 0 1 1 1 0 1 2 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 2 465 219 20 50 367 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1

10111100021 395 220 16 50 368 01111101122 00000011101

10111000011 303 220 18 50 368 01111011122 00000111111

110000000101 001110110021 466 220 22 50 368 000001110020 000000001101

1 1 1 0 0 0 0 0 1 2 1 3 0 0 0 1 1 0 0 0 0 1 472 P × MM3–11 224 12 51 375 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1

1110000000110 459 224 13 51 375 0001110000001 0000001111111

1 1 0 0 0 0 0 1 1 2 1 3 0 0 1 1 0 0 0 1 1 2 547 P × MM4–3 224 16 51 375 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 2

110000001111 1 3 001100000000 508 P × MM3–12 224 16 51 375 000011100011 000000011111

11000000010 401 224 16 51 375 00110000010 00001111112

1 1 1 0 0 0 0 1 2 363 224 19 51 375 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 1 1 1

11100111122 310 224 19 51 375 00010000101 00001111112

1 1 1 1 0 1 1 3 170 224 20 51 375 0 0 0 0 1 1 1 1

11000000111 00110000001 473 224 22 51 375 00001100001 00000010011 00000001111

1111000100111 296 225 16 51 376 0000111011111 0000000111011

1 0 0 1 0 1 1 1 2 361 225 19 51 376 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 2

110000111011 001100111011 383 226 18 51 377 000011000111 000000111111

1 1 0 1 0 0 0 0 0 1 0 0 1 0 1 1 0 0 0 1 379 226 20 51 377 0 0 0 1 1 1 0 1 1 2 0 0 0 0 0 0 1 1 1 1

11001000001 00110000001 398 226 23 51 377 00001011011 00000111001 00000000110

Continued on next page. FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 53

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

110011001011 001100110111 359 228 20 51 379 000011110011 000000001100

10000101111 165 229 19 52 383 01111101122 00000011111

1 0 0 0 0 0 1 0 1 1 0 1 1 0 0 0 0 0 0 1 297 229 21 52 383 0 0 0 1 1 1 1 0 1 2 0 0 0 0 0 0 0 1 1 1

1 1 1 0 0 0 2 0 2 28 230 13 51 381 0 0 0 1 1 1 0 2 2 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 0 1 1 1 0 0 1 1 1 0 1 1 0 2 403 230 18 52 384 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1

111000001011 000110000001 399 230 18 52 384 000001110111 000000001101

11100111122 302 230 18 52 384 00010000110 00001111112

11100000001101 00011100000011 384 230 18 52 384 00000011100110 00000000011001

1 1 1 0 0 0 0 2 0 2 0 0 0 1 1 0 0 0 0 1 271 230 19 52 384 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1

11110110122 166 230 20 52 384 00001110020 00000001101

11000000101 00110000101 471 230 22 52 384 00001100001 00000011011 00000000111

11000000011 00110010001 464 230 22 52 384 00001101001 00000011001 00000000111

1 0 1 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 4 0 0 1 1 0 1 0 1 0 1 499 BØS40 230 30 51 381 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1

1 1 1 0 0 1 0 0 1 2 0 0 0 1 1 0 1 0 0 1 294 235 19 53 392 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1

110011100011 001000100001 305 235 19 53 392 000111100011 000000011111

1 1 0 0 0 1 0 1 1 2 0 0 1 1 0 0 1 1 1 2 311 235 20 53 392 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 2

10000011001 01100000001 382 236 22 53 393 00011000001 00000111011 00000000110

1 1 0 0 0 0 1 1 2 394 238 22 54 398 0 0 1 1 1 0 1 1 2 0 0 0 0 0 1 1 1 2

Continued on next page. 54 T. COATES, A. KASPRZYK AND T. PRINCE

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

11100001111 381 239 13 54 399 00011100011 00000011111

1 0 0 0 0 0 1 0 169 240 −12 54 400 0 1 1 1 1 1 1 3

11000000000 1 3 502 P × MM2–23 240 8 54 400 00100000101 00011111122

1 1 0 0 0 0 0 0 0 1 3 402 P × MM2–24 240 12 54 400 0 0 1 1 1 0 0 0 2 0 0 0 0 0 1 1 1 1

11011000011 293 240 16 54 400 00111000011 00000111111

1 1 1 0 1 1 1 1 2 2 300 240 16 54 400 0 0 0 1 1 1 1 1 1 2

110000000010 001100000010 458 240 16 54 400 000011100011 000000011101

11000000000000 1 3 00111000000110 463 P × MM3–13 240 16 54 400 00000111000101 00000000111011

1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 0 1 380 240 19 54 400 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1

11000000011 00110000001 1 3 505 P × MM4–4 240 20 54 400 00001100000 00000011001 00000000111

1 1 0 0 0 0 0 0 1 0 2 0 0 1 1 0 0 0 0 0 1 548 F1 × S5 240 28 54 400 0 0 0 0 1 1 1 0 0 2 0 0 0 0 0 0 0 1 1 0

1 1 0 0 0 0 0 0 0 0 1 1 2 0 0 1 1 0 0 0 0 0 0 568 P × P × S5 240 28 54 400 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 2

1 1 1 1 0 2 2 4 70 240 32 55 403 0 0 0 0 1 1 1 2

11100001011 267 241 17 54 401 00011100111 00000011111

100001110011 011100000011 304 241 18 54 401 000011110111 000000001100

10110011112 358 242 14 54 402 01110000010 00001111112

1 0 0 0 0 0 0 1 0 1 38 242 30 55 405 0 1 1 1 1 1 1 2 2 3

1 0 0 0 0 0 1 1 1 1 36 244 16 55 407 0 1 1 1 1 1 1 1 2 2

1 1 0 0 0 0 1 1 1 2 0 0 1 1 0 0 1 1 1 2 417 244 24 55 407 0 0 0 0 1 0 0 1 1 2 0 0 0 0 0 1 1 1 1 2

110000000101 001110000111 393 245 19 55 408 000001110011 000000001101

1 1 0 1 0 1 1 0 1 2 0 0 1 0 1 1 1 0 0 2 392 246 18 55 409 0 0 0 1 1 1 1 0 1 2 0 0 0 0 0 0 0 1 1 0

Continued on next page. FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 55

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

110000000101 001000010010 378 246 18 55 409 000111110021 000000001101

1100000000101 0011000010010 457 246 21 55 409 0000111000011 0000000110010 0000000001101

1 1 1 0 0 0 0 2 0 2 0 0 0 1 1 0 0 0 1 1 223 248 16 55 411 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 0 0 1 144 249 17 56 415 0 0 1 0 0 0 0 1 1 0 0 0 1 1 1 1 1 2

1 0 1 0 1 0 0 1 0 0 1 1 0 0 1 0 0 1 4 31 BØS31 249 24 55 412 0 0 1 0 1 1 0 1 1 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1

11100000111 336 250 19 56 416 00011110112 00000001101

1 1 0 0 0 1 1 1 1 2 0 0 1 0 0 0 1 1 1 2 400 250 22 56 416 0 0 0 1 0 0 0 1 1 2 0 0 0 0 1 1 1 1 1 2

11000001011 00110000101 397 251 21 56 417 00001100001 00000011011 00000000111

1 0 1 1 0 0 1 1 2 262 252 16 56 418 0 1 1 1 0 1 1 1 2 0 0 0 0 1 1 1 1 2

11000000101 00110000101 460 252 20 56 418 00001100011 00000011011 00000000111

1 0 1 1 0 1 1 0 1 2 0 1 1 1 0 1 1 0 0 2 270 252 22 56 418 0 0 0 0 1 1 1 0 0 2 0 0 0 0 0 0 0 1 1 0

10000010000 01100000000 00101000001 2 2 504 S6 × S7 252 30 56 418 00011000000 00000110100 00000001100 00000000011

11110000212 108 254 18 57 423 00001110011 00000001101

1 1 1 0 0 0 1 1 2 279 255 19 57 424 0 0 0 1 0 0 0 1 1 0 0 0 0 1 1 1 1 2

1 1 0 0 0 0 0 0 1 1 3 388 P × MM2–25 256 8 57 425 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 2

1 1 1 0 0 0 0 1 1 224 256 12 57 425 0 0 0 1 1 1 0 0 2 0 0 0 0 0 0 1 1 0

11100001121 357 256 14 57 425 00011100001 00000011111

1110000000110 269 256 14 57 425 0001110000101 0000001111011

11000011111 163 256 15 57 425 00111000011 00000111111

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 0 0 1 0 1 1 2 301 256 16 57 425 0 0 1 1 0 1 1 1 2 0 0 0 0 1 1 1 1 2

1 1 0 0 0 0 0 1 0 1 1 3 0 0 1 1 0 0 0 0 0 0 456 P × MM3–15 256 16 57 425 0 0 0 0 1 1 1 0 1 2 0 0 0 0 0 0 0 1 1 1

11100001112 448 256 16 57 425 00011100010 00000011102

1 1 1 0 0 0 1 1 2 155 256 17 57 425 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 1 1

1 1 0 0 0 0 0 1 0 1 0 0 1 1 1 0 1 0 1 2 291 256 18 57 425 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 1 292 256 18 57 425 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1

111100000011 000101110111 360 256 18 57 425 000011110011 000000001100

111000010111 000111000011 268 256 18 57 425 000000110010 000000001101

10000110111 162 256 18 57 425 01111110122 00000001101

1 1 0 0 0 0 1 1 1 2 1 3 0 0 1 1 0 0 0 0 0 0 461 P × MM4–5 256 20 57 425 0 0 0 0 1 0 0 1 1 2 0 0 0 0 0 1 1 1 1 2

11000000011 00110000011 455 256 22 57 425 00001101001 00000011001 00000000111

10111001112 264 257 16 57 426 01111000011 00000111111

111000010011 000110101111 290 257 17 57 426 000001101111 000000011111

101100000010 011101100011 289 257 17 57 426 000011100001 000000011111

1 0 1 1 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 1 285 257 19 57 426 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1

1 1 0 0 1 1 0 0 2 343 258 10 57 427 0 0 1 1 1 1 0 1 2 0 0 0 0 0 0 1 1 0

110100000010 001011000001 353 258 18 57 427 000111001111 000000111111

11000110001 00100110101 375 258 21 57 427 00011001011 00000111001 00000000110

1 1 1 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1 377 260 16 58 432 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 0 0 0 0 0 1 1 0 2 164 260 16 58 432 0 1 1 1 1 1 1 1 2 2

10000001010 35 261 17 58 433 01000000101 00111111122

1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 1 1 0 1 2 376 261 20 58 433 0 0 0 0 1 1 1 0 1 2 0 0 0 0 0 0 0 1 1 1

1 1 0 1 0 1 0 1 1 2 0 0 1 0 1 1 0 0 0 1 288 262 18 58 434 0 0 0 1 1 1 0 1 1 2 0 0 0 0 0 0 1 1 1 1

111000000111 000111010011 266 262 19 58 434 000000110010 000000001101

1 0 0 0 0 1 0 1 1 142 266 16 59 441 0 1 1 1 1 1 0 0 2 0 0 0 0 0 0 1 1 1

1 1 1 0 0 0 0 0 2 2 0 0 0 1 1 0 1 0 0 1 261 266 19 59 441 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 1 1 1 2 0 0 1 1 0 0 1 1 1 2 309 266 20 59 441 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 2

110000001001 001110001011 369 267 17 59 442 000001110111 000000001101

1 1 0 0 1 1 1 0 0 2 0 0 1 0 1 1 1 0 1 2 387 268 16 59 443 0 0 0 1 1 1 1 0 0 2 0 0 0 0 0 0 0 1 1 0

1 1 1 0 0 2 2 2 4 342 268 22 60 446 0 0 0 1 0 0 1 1 2 0 0 0 0 1 1 1 1 2

1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 4 0 0 0 1 1 0 0 1 0 0 396 BØS64 268 24 59 443 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1

1 1 1 0 0 0 0 2 107 270 9 60 448 0 0 0 1 1 1 1 1

1 1 0 0 0 0 0 0 1 2 2 512 P × S5 270 21 60 448 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 1 2

1 1 1 0 0 0 1 0 1 2 0 0 0 1 1 0 0 0 1 1 284 271 19 60 449 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1

1 1 1 0 0 0 1 0 1 259 272 13 60 450 0 0 0 1 1 1 0 1 2 0 0 0 0 0 0 1 1 1

11000000101 352 272 14 60 450 00111110021 00000001101

11110002012 106 272 15 60 450 00001110111 00000001101

1 1 1 1 0 1 1 1 2 2 160 272 15 60 450 0 0 0 0 1 1 1 1 1 1

1 1 1 0 0 1 1 2 233 272 16 60 450 0 0 0 1 1 1 1 2

111000000111 1 3 000110000000 373 P × MM3–16 272 16 60 450 000001110011 000000001101

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 0 0 1 356 272 17 60 450 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1

11000000101 00110000000 1 3 454 P × MM4–7 272 20 60 450 00001100001 00000011011 00000000111

11001000001 00110101001 374 272 21 60 450 00001011011 00000111001 00000000110

11100011011 136 273 15 60 451 00011000111 00000111111

1 0 1 1 0 2 2 2 4 188 274 14 60 452 0 1 1 1 0 1 1 1 2 0 0 0 0 1 1 1 1 2

10011001111 220 274 16 60 452 01111000011 00000111111

1 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 0 0 1 260 277 17 61 458 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1

100010000101 011110000111 265 277 17 61 458 000001110011 000000001101

1 1 0 0 0 1 1 1 1 2 0 0 1 0 0 0 0 0 1 1 287 277 18 61 458 0 0 0 1 0 0 1 1 1 2 0 0 0 0 1 1 1 1 1 2

1 1 0 0 0 0 0 1 0 0 0 0 1 0 1 1 1 0 0 2 372 278 20 61 459 0 0 0 1 1 1 1 0 1 2 0 0 0 0 0 0 0 1 1 0

11000001001 00101100001 355 278 21 61 459 00011100011 00000011001 00000000110

1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 4 0 0 0 1 1 0 0 0 1 0 391 BØS63 278 24 61 459 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1

1 1 1 0 1 1 0 2 3 221 281 19 62 465 0 0 0 1 1 1 0 1 2 0 0 0 0 0 0 1 1 1

1 1 0 1 1 1 0 2 3 32 281 20 62 465 0 0 1 1 1 1 0 2 3 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 1 0 1 138 282 14 62 466 0 0 1 1 1 1 0 1 2 0 0 0 0 0 0 1 1 1

10000010101 161 282 16 62 466 01111110122 00000001101

1 1 0 1 1 0 1 1 2 141 282 18 62 466 0 0 1 1 1 0 1 1 2 0 0 0 0 0 1 1 1 2

111000000111 000111000111 349 282 18 62 466 000000110001 000000001101

1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 1 1 444 282 20 62 466 0 0 0 0 1 1 1 0 1 2 0 0 0 0 0 0 0 1 1 1

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

11000000011 00110000011 452 282 22 62 466 00001100011 00000011001 00000000111

1 1 0 0 0 0 1 0 1 1 0 0 1 1 1 0 0 1 1 2 154 283 18 62 467 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1

11000101011 00100011001 286 283 21 62 467 00011000001 00000111011 00000000110

1 1 1 0 0 0 0 2 0 2 0 0 0 1 1 0 1 0 1 1 133 284 16 62 468 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1

1 0 0 0 1 0 1 1 2 354 286 16 63 473 0 1 1 1 1 0 1 1 2 0 0 0 0 0 1 1 1 2

1 1 1 0 0 2 2 2 4 198 286 19 63 473 0 0 0 1 0 0 0 1 1 0 0 0 0 1 1 1 1 2

1 1 0 0 0 0 0 0 1 139 288 8 63 475 0 0 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 2

10000000101 335 288 13 63 475 01110000010 00001111112

1 1 0 0 0 0 0 0 1 247 288 15 63 475 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 1 1

10111000111 222 288 15 63 475 01111000011 00000111111

1 1 0 0 0 1 1 1 2 299 288 16 63 475 0 0 1 1 0 1 1 1 2 0 0 0 0 1 1 1 1 2

1 1 0 0 0 0 1 1 1 2 1 3 0 0 1 1 0 0 0 0 0 0 368 P × MM3–18 288 16 63 475 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 1 1 1 2

1 1 0 0 0 0 0 0 0 1 1 3 0 0 1 1 0 0 0 0 0 0 351 P × MM3–17 288 16 63 475 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1

110000000101 001110000011 348 288 16 63 475 000001110010 000000001101

1 0 0 0 1 0 1 0 0 1 0 1 1 1 0 1 1 0 1 2 282 288 17 63 475 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 1 1 0

11000100001 00100010000 371 288 20 63 475 00011010000 00000110111 00000001111

11001000001 00110000000 1 3 447 P × MM4–8 288 20 63 475 00001011011 00000111001 00000000110

1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 1 3 0 0 0 1 1 0 0 0 0 0 453 P × MM5–2 288 24 63 475 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 0 0 0 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 2 0 0 0 1 0 1 0 0 0 1 451 S6 × F1 288 24 63 475 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1

1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 2 1 1 0 0 0 1 1 0 0 0 0 0 501 S6 × P × P 288 24 63 475 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1

1 1 0 0 0 1 0 0 1 245 289 16 63 476 0 0 1 0 0 0 1 1 1 0 0 0 1 1 1 1 1 2

1 1 0 0 0 0 1 1 0 1 0 0 1 0 1 1 0 0 0 1 257 289 17 63 476 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1

1 0 1 1 0 1 1 0 1 2 0 1 1 1 0 0 0 0 1 1 132 289 18 63 476 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1

1 1 0 1 0 0 0 1 1 1 0 0 1 0 1 1 0 0 0 1 253 289 18 63 476 0 0 0 1 1 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1

1 1 0 0 0 1 0 1 0 1 0 0 1 1 0 0 1 0 1 1 249 290 18 63 477 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0

1 0 0 0 0 0 0 1 0 1 34 292 14 64 482 0 1 1 1 1 1 1 1 2 2

1 1 0 0 0 0 0 1 0 1 0 0 1 1 0 0 0 0 0 1 258 293 16 64 483 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 0 1 1 1 0 0 1 1 1 0 0 0 1 2 347 293 18 64 483 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1

11000000011 00100100011 367 293 21 64 483 00011100011 00000011001 00000000111

1 1 0 0 1 1 1 0 1 2 0 0 1 0 1 1 1 0 0 2 350 294 16 64 484 0 0 0 1 1 1 1 0 1 2 0 0 0 0 0 0 0 1 1 0

1 1 0 0 1 1 0 1 1 2 0 0 1 0 1 1 0 0 0 1 263 294 18 64 484 0 0 0 1 1 1 0 1 1 2 0 0 0 0 0 0 1 1 1 1

1 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 0 2 2 0 0 1 1 0 0 0 0 0 1 446 S7 × S7 294 25 64 484 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1

1 1 1 1 0 1 0 2 3 69 295 17 65 488 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 1

11000000001 140 296 16 65 489 00100000101 00011111222

1 1 0 0 0 0 0 1 30 297 13 65 490 0 0 1 1 1 1 1 2

1 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 4 0 0 0 1 1 1 0 0 0 0 390 BØS55 298 24 65 491 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 4 0 0 1 1 1 0 1 0 0 0 370 BØS57 298 24 65 491 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1

1 0 1 0 1 1 1 1 2 131 299 15 65 492 0 1 1 0 0 0 0 1 1 0 0 0 1 1 1 1 1 2

1 1 0 0 0 0 0 1 1 1 0 0 1 0 1 1 0 1 1 2 278 299 17 65 492 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1

1 0 0 1 1 1 1 0 0 2 0 1 0 0 0 0 1 0 0 1 137 299 17 65 492 0 0 1 1 1 1 1 0 1 2 0 0 0 0 0 0 0 1 1 0

1 1 0 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 4 283 BØS84 299 20 65 492 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1

1 1 1 0 0 0 0 0 2 2 0 0 0 1 1 0 0 0 1 1 254 302 19 66 498 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1

1 1 1 0 0 0 0 2 2 232 302 19 66 498 0 0 0 1 1 1 0 1 2 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 0 1 1 150 302 20 66 498 0 0 1 1 1 1 0 2 3 0 0 0 0 0 0 1 1 1

11110100011 104 303 13 66 499 00001101111 00000011111

1 1 1 0 0 1 1 0 1 2 0 0 0 1 0 0 1 0 0 1 218 303 17 66 499 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1

11000000000 1 3 277 P × MM2–27 304 12 66 500 00111000011 00000111111

1 0 0 0 0 0 1 1 0 1 159 304 12 66 500 0 1 1 1 1 1 1 1 2 2

11100001111 134 304 13 66 500 00011100011 00000011101

1 1 0 0 1 1 1 1 2 248 304 14 66 500 0 0 1 0 0 1 1 1 2 0 0 0 1 1 1 1 1 2

1 1 0 0 0 1 0 1 1 152 304 14 66 500 0 0 1 1 1 0 1 1 2 0 0 0 0 0 1 1 1 1

1 1 1 0 2 2 2 4 182 304 16 66 500 0 0 0 1 1 1 1 2

1 0 0 0 0 1 1 1 0 1 0 1 1 1 0 0 0 0 1 1 252 304 16 66 500 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0

1 1 0 0 0 0 0 0 1 1 1 3 0 0 1 1 1 0 0 0 1 2 443 P × MM3–21 304 16 66 500 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1

1 0 0 0 1 1 0 0 0 1 1 3 0 1 1 0 0 0 0 0 0 0 346 P × MM3–20 304 16 66 500 0 0 0 1 1 1 0 1 1 2 0 0 0 0 0 0 1 1 1 1

1 1 1 0 0 0 0 0 1 1 0 0 0 1 1 0 0 0 0 1 219 304 16 66 500 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 1 0 0 0 1 1 2 1 3 256 P × MM3–19 304 16 66 500 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 2

1 0 1 1 0 1 0 0 1 1 0 1 0 0 1 1 0 0 0 1 255 304 17 66 500 0 0 1 1 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1

1 1 0 0 0 0 1 0 1 1 0 0 1 1 0 0 0 1 1 1 153 304 17 66 500 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1

1 1 0 0 0 0 0 0 1 1 0 0 1 1 0 0 0 0 1 1 334 304 18 66 500 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1

1 0 0 0 1 1 1 0 0 2 1 3 0 1 1 0 0 0 0 0 0 0 445 P × MM4–9 304 20 66 500 0 0 0 1 1 1 1 0 1 2 0 0 0 0 0 0 0 1 1 0

11100010011 123 305 13 66 501 00011101111 00000011111

1 1 0 1 0 0 0 0 1 0 0 1 0 1 0 0 0 0 4 333 BØS33 305 21 66 501 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1

1 1 0 0 0 1 0 0 0 1 0 0 1 1 0 0 1 0 0 1 331 306 18 66 502 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 1 1 0

1 0 0 1 0 0 0 0 0 0 1 0 0 1 0 0 0 0 4 441 BØS39 307 23 66 503 0 0 1 1 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1

1 0 0 1 1 0 0 0 2 2 0 1 1 1 1 0 0 0 2 2 105 308 18 67 507 0 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 1 0 0 0 0 0 1 1 0 0 1 0 0 0 4 0 0 0 0 1 1 0 0 0 0 389 BØS49 308 24 67 507 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1

1 1 0 0 1 1 1 0 1 2 0 0 1 0 0 0 1 0 1 1 135 309 18 67 508 0 0 0 1 1 1 1 0 1 2 0 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 1 246 309 19 67 508 0 0 0 1 1 1 1 0 1 2 0 0 0 0 0 0 0 1 1 1

1 0 1 1 0 0 0 0 1 1 0 1 1 1 0 0 0 1 1 1 130 310 17 67 509 0 0 0 0 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1

1 1 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 4 281 BØS69 310 20 67 509 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1

1 0 0 0 0 1 1 0 1 0 1 0 1 0 0 0 1 1 4 217 BØS29 310 21 67 509 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1

1 1 1 0 0 0 1 0 0 1 0 0 0 1 1 0 0 1 0 1 244 314 16 68 516 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 1 0 0 1 1 1 2 215 319 13 69 524 0 0 0 1 0 0 0 1 1 0 0 0 0 1 1 1 1 2

1 0 0 0 0 0 0 1 0 0 158 320 0 69 525 0 1 1 1 1 1 1 1 2 2

1 1 1 1 0 0 0 2 3 1 3 149 P × MM2–28 320 8 69 525 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1

1 1 0 0 0 1 1 1 2 1 3 341 P × MM2–29 320 12 69 525 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1 2

1 1 1 0 0 0 0 0 2 2 1 3 0 0 0 1 1 0 0 0 0 1 216 P × MM3–22 320 16 69 525 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1

1 1 1 0 0 0 0 2 2 115 320 16 69 525 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 1 1 1

1 1 1 0 1 1 0 1 2 187 320 16 69 525 0 0 0 1 1 1 0 0 2 0 0 0 0 0 0 1 1 0

1 0 1 1 0 0 0 0 0 1 0 1 1 1 0 0 0 0 1 1 332 320 16 69 525 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0

1 0 1 1 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 1 243 320 17 69 525 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 1 1 1

1 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 3 345 P × MM4–10 320 20 69 525 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1

1 1 0 0 0 0 1 0 0 1 0 0 1 1 0 0 0 0 0 1 238 321 17 69 526 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 1 1 0

1 1 0 0 0 0 0 1 1 1 0 0 1 0 0 1 0 1 1 1 242 321 17 69 526 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1

1 1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0 1 4 241 BØS28 321 21 69 526 0 0 0 1 0 1 0 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1

1 0 1 0 1 1 0 1 4 0 1 0 1 1 0 1 1 25 BØS32 322 18 69 527 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 0 0 1 102 324 12 70 532 0 0 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1

1 0 0 0 1 0 0 0 0 0 1 1 1 0 0 0 0 0 2 2 365 S6 × P 324 18 70 532 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1

1 1 0 0 0 0 1 0 1 0 0 1 0 1 0 0 0 0 4 251 BØS80 325 20 70 533 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1

1 0 1 1 1 0 0 0 2 214 326 16 70 534 0 1 1 1 1 0 1 1 2 0 0 0 0 0 1 1 1 0

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 1 0 0 0 0 0 4 0 0 1 0 1 1 0 1 129 BØS42 326 17 70 534 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1

1 0 1 0 1 0 1 0 4 0 1 0 1 1 0 0 1 27 BØS30 327 18 70 535 0 0 1 1 1 0 1 1 0 0 0 0 0 1 1 1

1 1 1 0 0 0 1 0 2 2 0 0 0 1 1 0 0 0 0 1 210 329 17 71 540 0 0 0 0 0 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1

1 1 0 0 1 1 1 1 2 29 330 14 71 541 0 0 1 0 0 0 0 1 1 0 0 0 1 1 1 1 1 2

1 1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 4 280 BØS53 330 20 71 541 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1

1 1 0 0 0 0 0 0 1 0 0 0 1 0 1 1 1 0 1 2 366 330 20 71 541 0 0 0 1 1 1 1 0 0 2 0 0 0 0 0 0 0 1 1 0

1 0 0 0 0 1 1 0 1 1 0 1 1 1 0 1 1 0 0 2 275 331 16 71 542 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 0

1 1 0 0 0 0 0 0 1 1 0 0 1 0 0 0 1 0 1 1 209 331 17 71 542 0 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 4 250 BØS65 331 20 71 542 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1

1 0 0 0 1 0 1 0 1 0 1 0 0 0 0 1 0 0 4 237 BØS66 332 21 71 543 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1

1 0 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 4 344 BØS96 334 18 72 548 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1

1 1 1 0 0 0 0 1 1 231 335 13 72 549 0 0 0 1 1 1 0 1 2 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 0 1 1 122 335 15 72 549 0 0 1 1 1 1 0 1 2 0 0 0 0 0 0 1 1 1

1 1 1 0 0 0 1 0 0 1 0 0 0 1 0 1 0 0 0 0 239 335 16 72 549 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 1 1 1

1 0 0 0 1 1 0 2 2 19 335 18 72 549 0 1 1 1 1 1 0 2 3 0 0 0 0 0 0 1 1 1

10000000100 103 336 12 72 550 01110000011 00001111111

11100000011 230 336 12 72 550 00011100010 00000011101

1 0 1 1 1 0 0 0 1 101 336 13 72 550 0 1 1 1 1 0 1 1 2 0 0 0 0 0 1 1 1 1

1 1 1 0 1 1 0 1 2 99 336 14 72 550 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 1 1 1

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 0 1 1 0 1 1 2 126 336 14 72 550 0 0 1 1 1 0 1 1 2 0 0 0 0 0 1 1 1 1

1 0 1 1 0 0 0 0 1 1 1 3 0 1 1 1 0 0 0 0 0 1 330 P × MM3–24 336 16 72 550 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1

1 0 0 0 0 0 1 0 1 1 1 3 0 1 1 0 0 0 0 0 0 0 240 P × MM3–23 336 16 72 550 0 0 0 1 1 1 1 0 1 2 0 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 2 340 S7 × F1 336 20 72 550 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1

1 0 0 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 2 1 1 442 S7 × P × P 336 20 72 550 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1

1 0 1 1 0 0 2 0 4 0 1 1 1 0 0 1 0 90 BØS16 337 16 72 551 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1

1 1 1 0 0 0 2 0 2 22 338 13 72 552 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 1

1 1 1 1 0 0 0 3 3 59 341 17 74 561 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0 0 0 4 276 BØS59 341 20 73 558 0 0 0 1 0 1 1 0 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 1 1

1 1 1 1 1 0 3 4 6 341 29 74 561 0 0 0 0 0 1 1 1

1 0 0 0 0 1 0 1 1 68 345 17 74 565 0 1 1 1 1 1 0 2 3 0 0 0 0 0 0 1 1 1

1 0 1 1 1 1 0 0 2 128 346 12 74 566 0 1 1 1 1 1 0 1 2 0 0 0 0 0 0 1 1 0

1 1 0 0 0 0 1 0 4 0 0 1 0 1 1 0 0 121 BØS93 347 16 74 567 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1

1 1 1 1 0 1 1 2 67 350 12 75 573 0 0 0 0 1 1 1 2

1 1 1 0 0 0 0 0 1 1 18 351 9 75 574 0 0 0 1 1 1 1 1 1 1

1 1 1 1 0 0 0 0 4 0 0 0 1 0 1 0 1 206 BØS44 351 15 75 574 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1

1 1 1 1 0 0 1 2 114 352 12 75 575 0 0 0 0 1 1 1 1

1 1 1 0 0 0 1 0 1 125 352 12 75 575 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 0

1 1 1 0 0 0 0 1 1 119 352 13 75 575 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 1 1

1 1 0 0 0 0 1 1 1 207 352 13 75 575 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 1 1

1 0 0 0 0 0 1 1 1 66 352 14 75 575 0 1 0 1 1 1 2 2 3 0 0 1 1 1 1 1 1 2

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 0 0 0 0 1 0 4 0 0 1 1 0 0 1 0 151 BØS73 352 16 75 575 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1

1 1 0 0 0 1 0 1 4 0 0 1 1 0 0 1 0 127 BØS85 352 16 75 575 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1

1 1 0 0 0 0 1 0 1 3 0 0 1 1 0 0 0 0 236 P × MM3–25 352 16 75 575 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1

1 1 0 0 0 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 208 352 16 75 575 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 1 3 339 P × MM4–12 352 20 75 575 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1

1 0 0 1 1 0 0 1 1 95 353 14 75 576 0 1 1 1 1 0 1 1 2 0 0 0 0 0 1 1 1 1

1 1 0 0 1 0 0 1 1 203 353 14 75 576 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 1 1

1 0 0 0 1 1 0 2 2 17 353 15 75 576 0 1 1 1 1 1 0 1 2 0 0 0 0 0 0 1 1 1

1 1 1 0 1 1 0 0 0 0 0 0 1 1 0 0 0 0 4 338 BØS23 354 18 76 580 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1

1 1 0 1 0 1 0 1 4 0 0 1 0 1 1 0 0 100 BØS81 357 16 76 583 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1

1 0 0 0 1 0 0 1 1 1 0 1 1 1 1 0 0 1 1 2 98 357 17 76 583 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1

1 0 0 0 0 1 0 0 0 0 0 1 1 1 0 1 0 0 0 1 229 362 16 77 591 0 0 0 0 1 1 0 1 1 1 0 0 0 0 0 0 1 1 1 1

1 1 0 0 0 0 0 1 1 1 0 0 1 1 1 0 0 1 1 2 148 362 17 77 591 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 1 1 1

1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 1 4 274 BØS68 363 20 77 592 0 0 0 1 1 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1

1 0 0 1 1 0 0 0 4 0 1 0 1 1 0 0 1 24 BØS87 364 17 77 593 0 0 1 1 1 0 1 1 0 0 0 0 0 1 1 1

1 1 1 0 0 2 0 0 0 0 0 0 1 1 0 0 0 0 4 337 BØS4 364 18 78 596 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1

1 0 0 0 0 1 0 0 4 0 1 1 1 0 0 1 0 205 BØS102 367 15 78 599 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1

1 1 0 0 1 1 1 1 2 124 368 12 78 600 0 0 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1 2

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 1 0 0 0 1 1 2 1 3 228 P × MM2–31 368 12 78 600 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1 1

1 1 0 0 0 0 0 0 0 1 3 120 P × MM2–30 368 12 78 600 0 0 1 0 0 0 0 1 1 0 0 0 1 1 1 1 1 2

1 1 1 0 0 0 0 1 1 113 368 13 78 600 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 1 1 1

1 0 0 0 1 1 0 0 1 3 0 1 1 0 0 0 0 0 204 P × MM3–26 368 16 78 600 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1

1 0 0 0 1 1 0 0 1 1 0 1 1 0 1 1 0 0 1 1 213 368 16 78 600 0 0 0 1 1 1 0 0 0 1 0 0 0 0 0 0 1 1 1 1

1 0 0 0 0 1 1 0 0 1 0 1 1 1 0 1 1 0 1 2 212 368 16 78 600 0 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 0 1 1 0

1 1 0 1 0 0 0 1 4 0 0 1 0 1 1 0 1 97 BØS13 368 17 78 600 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1

1 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 1 3 329 P × MM4–13 368 20 78 600 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1

1 0 1 1 0 0 1 1 1 88 369 13 78 601 0 1 1 1 0 0 0 0 1 0 0 0 0 1 1 1 1 1

1 1 0 0 1 0 0 1 4 0 0 1 0 0 1 0 0 201 BØS34 369 17 78 601 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1

1 1 0 0 1 0 0 0 4 0 0 1 0 0 0 0 1 118 BØS35 369 17 78 601 0 0 0 1 1 0 1 1 0 0 0 0 0 1 1 1

1 0 0 1 1 0 2 2 2 62 370 14 78 602 0 1 1 1 1 0 1 1 2 0 0 0 0 0 1 1 1 1

1 0 0 0 0 0 1 0 1 0 1 0 0 1 0 1 0 0 4 235 BØS58 373 20 79 608 0 0 1 1 1 0 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1

1 1 1 1 0 0 0 2 2 58 376 12 80 614 0 0 0 0 1 1 0 0 1 0 0 0 0 0 0 1 1 1

1 0 0 0 0 0 1 1 5 376 14 80 614 0 1 1 1 1 1 2 3

1 1 0 1 1 1 1 2 26 378 10 80 616 0 0 1 1 1 1 1 2

1 1 0 0 0 0 0 1 1 89 378 12 80 616 0 0 1 1 1 1 0 0 1 0 0 0 0 0 0 1 1 1

1 0 0 0 0 1 0 0 2 2 0 1 1 1 0 0 0 0 227 S7 × P 378 15 80 616 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1

1 1 1 0 0 0 2 0 4 0 0 0 1 0 1 0 0 193 BØS9 382 15 81 623 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1

1 0 1 1 1 0 1 2 2 65 383 13 81 624 0 1 1 1 1 0 1 1 2 0 0 0 0 0 1 1 1 1

Continued on next page. 68 T. COATES, A. KASPRZYK AND T. PRINCE

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 0 0 0 0 1 1 1 147 384 13 81 625 0 0 1 1 1 0 1 1 2 0 0 0 0 0 1 1 1 1

1 0 0 0 1 0 1 1 1 93 384 13 81 625 0 1 1 1 1 0 1 1 2 0 0 0 0 0 1 1 1 1

1 1 1 0 0 1 1 1 2 16 384 13 81 625 0 0 0 1 0 0 0 1 1 0 0 0 0 1 1 1 1 1

1 0 1 1 0 1 0 1 4 0 1 1 1 0 0 0 1 185 BØS36 384 16 81 625 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1

1 1 0 0 0 0 0 1 0 0 1 1 0 1 0 0 226 F1 × F1 384 16 81 625 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1

1 1 0 0 0 1 0 1 4 0 0 1 1 0 0 1 1 96 BØS91 384 16 81 625 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 1

1 0 0 0 0 1 0 1 4 0 1 0 1 1 1 0 1 86 BØS92 384 16 81 625 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1

1 1 0 0 0 0 0 1 1 1 0 0 1 1 0 0 0 0 328 P × P × F1 384 16 81 625 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1

1 0 0 0 1 0 0 0 4 0 1 0 0 0 1 0 0 324 BØS38 385 17 81 626 0 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1

1 1 0 0 0 0 0 1 4 0 0 1 0 1 1 0 1 117 BØS88 389 16 82 633 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1

1 1 0 0 1 1 0 0 4 0 0 1 0 1 1 0 1 94 BØS71 390 16 82 634 0 0 0 1 1 1 0 0 0 0 0 0 0 0 1 1

1 1 0 0 0 0 1 0 0 0 0 1 1 0 0 1 0 0 4 273 BØS50 394 20 83 641 0 0 0 0 1 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1

1 1 1 0 0 2 0 4 75 BØS20 400 12 84 650 0 0 0 1 1 0 1 0 0 0 0 0 1 1

1 1 0 0 0 1 0 4 85 BØS113 400 12 84 650 0 0 1 1 1 0 1 0 0 0 0 0 1 1

1 1 1 1 0 1 0 1 2 64 400 12 84 650 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 1 1

1 1 1 0 0 0 0 1 1 197 400 12 84 650 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 1 1 0

1 0 1 1 0 0 0 2 1 3 0 1 1 1 0 0 0 1 184 P × MM3–29 400 16 84 650 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1

1 1 0 0 0 0 1 1 1 3 0 0 1 1 0 0 0 0 200 P × MM3–30 400 16 84 650 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 1

1 0 1 1 0 0 0 4 21 BØS114 401 13 84 651 0 1 1 1 0 1 1 0 0 0 0 1 1 1

1 1 0 0 1 0 1 4 15 BØS86 405 12 85 658 0 0 1 1 0 1 1 0 0 0 0 1 1 1

Continued on next page. FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 69

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 0 0 0 0 1 0 1 0 0 1 0 0 0 1 0 0 4 234 BØS54 405 20 85 658 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1

1 0 0 0 0 0 1 0 0 0 1 0 0 1 0 1 0 1 4 202 BØS56 405 20 85 658 0 0 1 1 1 0 1 0 1 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1

1 0 1 1 0 0 1 4 23 BØS46 406 13 85 659 0 1 1 1 0 0 0 0 0 0 0 1 1 1

1 1 1 0 1 0 0 1 4 0 0 0 1 0 1 0 1 196 BØS25 409 15 86 665 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1

1 1 0 0 1 1 0 1 4 0 0 1 0 1 1 0 0 87 BØS70 411 16 86 667 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1

1 0 0 0 0 1 0 0 4 0 1 1 1 0 1 0 0 199 BØS100 415 15 87 674 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1

1 0 1 1 0 1 0 2 4 0 1 0 0 1 1 0 1 57 BØS11 415 16 87 674 0 0 1 1 1 1 0 1 0 0 0 0 0 0 1 1

1 1 1 0 0 1 1 1 2 92 416 12 87 675 0 0 0 1 0 0 0 1 0 0 0 0 0 1 1 1 1 1

1 1 0 0 0 0 0 1 1 3 0 0 1 1 0 0 0 1 225 P × MM3–31 416 16 87 675 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1

1 0 1 1 0 0 1 4 12 BØS37 417 13 87 676 0 1 1 1 0 1 1 0 0 0 0 1 1 1

1 1 0 0 1 0 1 1 4 0 0 1 0 0 0 0 1 112 BØS27 417 17 87 676 0 0 0 1 1 0 0 1 0 0 0 0 0 1 1 1

1 0 0 0 0 0 0 1 0 63 430 12 90 698 0 1 0 1 1 1 1 2 2 0 0 1 1 1 1 1 1 2

1 0 0 0 0 0 1 1 4 431 9 90 699 0 1 1 1 1 1 1 2

1 0 1 1 1 0 0 4 55 BØS104 431 11 90 699 0 1 1 1 1 0 1 0 0 0 0 0 1 1

1 3 1 1 0 0 0 0 0 0 111 P × Q 432 8 90 700 0 0 1 1 1 1 1 2

1 1 1 0 1 1 1 2 14 432 9 90 700 0 0 0 1 1 1 1 1

1 1 1 0 0 0 0 1 78 432 9 90 700 0 0 0 1 1 1 1 1

1 1 1 0 1 0 0 4 80 BØS45 432 12 90 700 0 0 0 1 1 0 1 0 0 0 0 0 1 1

1 1 0 0 0 1 1 1 3 83 P × MM2–33 432 12 90 700 0 0 1 1 0 0 0 0 0 0 0 1 1 1

1 1 0 0 0 0 1 2 110 P × F1 432 12 90 700 0 0 1 1 1 0 0 0 0 0 0 0 1 1

1 1 0 0 0 0 0 2 1 1 195 P × P × P 432 12 90 700 0 0 1 1 0 0 0 0 0 0 0 1 1 1

Continued on next page. 70 T. COATES, A. KASPRZYK AND T. PRINCE

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 1 0 0 0 1 1 1 4 0 0 1 0 0 0 1 1 84 BØS82 432 16 90 700 0 0 0 1 0 0 0 1 0 0 0 0 1 1 1 1

1 1 0 0 1 0 0 4 81 BØS41 433 13 90 701 0 0 1 1 1 0 1 0 0 0 0 0 1 1

1 0 0 0 1 1 0 2 4 0 1 1 0 1 1 0 1 56 BØS15 433 16 90 701 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1

1 0 1 1 0 1 0 2 4 0 1 1 1 0 1 0 1 77 BØS24 442 16 92 716 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1

1 1 0 0 0 0 1 0 1 0 0 1 1 0 0 1 0 1 4 211 BØS48 442 20 92 716 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 1 1

1 0 0 0 0 1 0 1 1 50 446 14 93 723 0 1 1 1 1 2 0 2 3 0 0 0 0 0 0 1 1 1

1 0 0 0 0 1 0 0 4 0 1 1 1 0 1 0 1 194 BØS95 447 15 93 724 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1

1 1 0 0 0 0 0 1 4 0 0 1 0 0 1 0 1 116 BØS60 448 16 93 725 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1

1 0 1 1 0 2 2 4 9 BØS17 450 13 93 727 0 1 1 1 0 1 1 0 0 0 0 1 1 1

1 1 0 0 1 1 1 4 11 BØS94 459 12 95 742 0 0 1 0 0 1 1 0 0 0 1 1 1 1

1 0 1 1 0 0 0 2 4 0 1 1 1 0 0 0 1 82 BØS6 463 16 96 749 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1

1 1 1 1 0 2 2 3 49 464 11 96 750 0 0 0 0 1 1 1 1

1 1 1 0 1 0 1 4 181 BØS43 464 12 96 750 0 0 0 1 1 0 0 0 0 0 0 0 1 1

1 1 0 0 1 1 1 4 54 BØS109 464 12 96 750 0 0 1 0 0 0 1 0 0 0 1 1 1 1

1 1 0 0 0 0 1 1 4 0 0 1 1 0 0 1 1 91 BØS52 464 16 96 750 0 0 0 0 1 0 0 1 0 0 0 0 0 1 1 1

1 1 1 0 0 2 0 0 4 0 0 0 1 0 1 0 1 192 BØS5 478 15 99 773 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1

1 0 0 0 1 0 1 4 76 BØS111 480 12 99 775 0 1 1 1 1 0 1 0 0 0 0 0 1 1

1 1 1 0 0 0 0 1 1 109 480 12 99 775 0 0 0 1 1 1 0 1 1 0 0 0 0 0 0 1 1 0

1 1 0 0 0 0 0 1 4 0 0 1 1 0 0 0 1 146 BØS51 480 16 99 775 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1

1 1 0 0 0 1 1 4 13 BØS74 486 12 100 784 0 0 1 1 0 1 1 0 0 0 0 1 1 1

Continued on next page. FOUR-DIMENSIONAL FANO TORIC COMPLETE INTERSECTIONS 71

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 0 0 0 0 0 1 4 48 BØS105 489 11 101 790 0 1 0 1 1 1 2 0 0 1 1 1 1 1

1 1 1 0 0 0 2 1 3 180 P × MM2–36 496 12 102 800 0 0 0 1 1 0 0 0 0 0 0 0 1 1

1 1 0 0 0 0 1 4 79 BØS106 496 12 102 800 0 0 1 1 1 0 1 0 0 0 0 0 1 1

1 0 0 0 1 1 0 2 4 0 1 1 0 1 1 0 2 53 BØS10 496 16 102 800 0 0 0 1 1 1 0 1 0 0 0 0 0 0 1 1

1 1 1 0 0 2 0 1 4 0 0 0 1 0 1 0 0 191 BØS22 505 15 104 815 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1

4 1 1 0 1 1 1 2 BØS115 512 8 105 825 0 0 1 1 1 1

4 1 1 1 0 0 1 10 BØS47 513 9 105 826 0 0 0 1 1 1

4 1 1 1 0 1 1 8 BØS118 513 9 105 826 0 0 0 1 1 1

1 1 0 0 1 0 2 4 52 BØS18 529 13 108 851 0 0 1 1 1 0 1 0 0 0 0 0 1 1

4 1 0 0 0 0 1 47 BØS121 544 8 111 875 0 1 1 1 1 1

1 1 1 0 0 2 0 2 4 0 0 0 1 0 1 0 0 190 BØS3 558 15 114 898 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1 1

1 0 0 1 1 0 2 4 45 BØS12 560 12 114 900 0 1 1 1 1 0 2 0 0 0 0 0 1 1

1 1 1 0 1 0 2 4 73 BØS26 560 12 114 900 0 0 0 1 1 0 1 0 0 0 0 0 1 1

1 1 1 0 0 2 2 4 72 BØS8 576 12 117 925 0 0 0 1 0 0 1 0 0 0 0 1 1 1

1 1 1 0 0 0 2 4 74 BØS7 592 12 120 950 0 0 0 1 1 0 1 0 0 0 0 0 1 1

4 1 1 1 0 2 2 7 BØS21 594 9 120 952 0 0 0 1 1 1

1 0 1 1 1 0 3 4 44 BØS1 605 11 123 972 0 1 1 1 1 0 2 0 0 0 0 0 1 1

1 0 0 0 0 0 1 0 46 624 8 126 1000 0 1 1 1 1 1 2 2

4 1 1 1 1 0 3 42 BØS2 800 8 159 1275 0 0 0 0 1 1

Table 4: Certain 4-dimensional Fano manifolds with Fano index r = 2 that arise as complete intersections in toric Fano manifolds.

4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

4 682 V4 64 188 21 125 1 1 1 1 1 1 4 4 640 V6 96 90 27 175 1 1 1 1 1 1 1 2 3 4 596 V8 128 48 33 225 11111111222

1 3 1 1 0 0 0 0 0 0 567 P × B3 192 −12 45 325 0 0 1 1 1 1 1 3

Continued on next page. 72 T. COATES, A. KASPRZYK AND T. PRINCE

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4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

1 3 1 1 0 0 0 0 0 0 0 0 500 P × B4 256 0 57 425 0 0 1 1 1 1 1 1 2 2

4 1 1 1 0 0 0 0 1 327 MW5 256 16 57 425 0 0 0 1 1 1 1 2

4 1 1 1 0 0 0 0 0 1 0 326MW 8 320 10 69 525 0 0 0 1 1 1 1 1 1 2

4 1 1 1 1 0 0 0 0 1 1 189 MW7 320 12 69 525 0 0 0 0 1 1 1 1 1 1

4 1 1 1 0 1 1 1 2 186MW 10 352 10 75 575 0 0 0 1 1 1 1 2

1 1 0 0 0 0 0 0 0 1 3 325 P × MM2–32 384 12 81 625 0 0 1 1 1 0 0 0 1 0 0 0 0 0 1 1 1 1

1 1 0 0 0 0 0 0 1 1 1 1 0 0 1 1 0 0 0 0 440 P × P × P × P 384 16 81 625 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 1

4 1 1 1 1 0 1 1 2 61 MW12 416 10 87 675 0 0 0 0 1 1 1 1

1 1 1 0 0 0 1 1 3 183 P × MM2–35 448 12 93 725 0 0 0 1 1 0 0 0 0 0 0 0 1 1

4 1 0 0 0 0 0 1 0 60 MW13 480 8 99 775 0 1 1 1 1 1 1 2

1 3 1 1 0 0 0 0 51 P × P 512 8 105 825 0 0 1 1 1 1

4 1 1 1 1 0 2 43 MW15 640 8 129 1025 0 0 0 0 1 1

Table 5: Certain 4-dimensional Fano manifolds with Fano index r = 3 that arise as complete intersections in toric Fano manifolds.

4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

4 37FI 3 243 27 55 406 1 1 1 1 1 1 3 4 33 FI4 324 12 70 532 1 1 1 1 1 1 1 2 2

2 2 1 1 1 0 0 0 20 P × P 486 9 100 784 0 0 0 1 1 1

Table 6: The 4-dimensional Fano manifold with Fano index r = 4 as a hypersurface in a toric Fano manifold.

4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

4 3 Q 512 6 105 825 1 1 1 1 1 1 2

Table 7: The 4-dimensional Fano manifold with Fano index r = 5 is toric.

4 0 0 Period ID Description (−KX ) χ(TX ) h (−KX ) h (−2KX ) Weights and line bundles

4 1 P 625 5 126 1001 1 1 1 1 1

Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK E-mail address: [email protected] E-mail address: [email protected] E-mail address: [email protected]