The graded ring database for Fano 3-folds and the Bogomolov stability bound

Abstract

My paper (Suzuki 2003) produced some computer routines in Magma (Bosma et al. J Symb Comp 24:235–265, 1997) for the numerical invariants of Fano 3-folds, and used them in particular to determine the maximum value $f=19$ of the Fano index. As a byproduct of the research, extensive data associated with all possible sets of singular points of Fano 3-folds with Fano indices greater than or equal to 2 was obtained. Collaborative research with Gavin Brown developed an improved version of the Magma program. The data discussed above was added to the Graded Ring Data Base (Brown et al. 2015) at the University of Kent. Subsequently, GRDB, now located to the University of Warwick, recently modified its interface to accommodate additional conditions, facilitating a more refined selection of Fano manifolds. In this context, we focus on the inequality known as the Bogomolov stability bound. We present a list of candidates for Fano 3-folds that do not satisfy these conditions and propose the conjecture that they do not exist.This result has been independently obtained in Liu and Liu (2023).

1 Symbols

In the following, we work over $\mathbb C$.

Definition 1

Fano 3-folds are 3-dimensional normal projective algebraic varieties with Picard number 1, having at most terminal singularities, and satisfying the conditions that the anti-canonical divisor $-K_X$ is an ample $\mathbb {Q}$-Cartier divisor.

Let (U, P) be a germ of a 3-dimensional terminal singularity of index $r = r_P > 1$. According to [16] and [13], if (U, P) is not a quotient singular point, it can be deformed to a unique collection of a finite number of terminal quotient singularities, say $\{{P_k }\}^{n_P}_{k=1}$. Write a type of singularity of $P_k$ as

$$\begin{aligned}\displaystyle \frac{1}{r_{P,k}} \left( f, a_{P,k}, -a_{P,k}\right) \hspace{0.5cm} \text {or} \hspace{0.5cm} [r_{P,k}, a_{P,k}].\end{aligned}$$

Then $(r_{P,k}, a_{P,k}) = 1$ and $r_k \ge 2, n_P \ge 2$ and $r_P = {{\,\textrm{lcm}\,}}(r_{P,k})$ hold. We call the set ${\mathcal {B}}(U, P):= {\{P_k\} }^{n_P}_{k=1}$ the basket of singularities of (U, P). When (U, P) is already a quotient singular point, we regard $\{(U, P )\}$ itself as the basket of singularities of (U, P). Since the 3-dimensional terminal singularities are isolated singularities, we can speak of the basket of singularities in the global case:

Definition 2

Let X be a terminal 3-fold. Let $\{Pi\}^m_{i=1}$ be the set of singular points of X with $r(P_i) \ge 2$ and $P_i \in U_i$ be a small analytic neighborhood of $P_i$. Then, we call the disjoint union $\cup ^m_{i=1}{\mathcal {B}}(Ui, Pi)$ the basket of singularities of X.

Definition 3

For Fano 3-fold X, the Fano index of X is defined as follows:

$$\begin{aligned} f=f(X):= \max \bigg \{n \in \mathbb Z_{> 0} {\ \vert \ }-K_X \equiv nA_X \hbox { for}\,\, A_X \,\,\hbox { a Weil divisor} \bigg \} \end{aligned}$$

where $\equiv$ denotes linear equivalence of some multiple.

We call the above $A_X$ as a primitive ample divisor.

Definition 4

The Hilbert series of Fano 3-fold X with primitive ample divisor A is the formal power series in t given by $\sum \dim H^0(X,{\mathcal {O}}_X(nA))t^n$. Moreover, we define the graded ring

$$\begin{aligned} {{\,\textrm{Proj}\,}}\bigoplus _{n\ge 0}H^0 \bigg (X,{\mathcal {O}}_X(nA)\bigg ) \end{aligned}$$

as the sections ring of A on X.

This definition establishes a one-to-one correspondence between Fano varieties and their section rings. Therefore, to consider the existence of a Fano variety, it suffices to show the existence of the corresponding sections ring. Specifically, generalized singular Riemann–Roch theorem [18, 19], Hilbert series

$$\begin{aligned} P_X: = \sum _{n \ge 0}\dim H^0 \bigg (X_n,{\mathcal {O}}(-nA)\bigg ) t^n \end{aligned}$$

can be explicitly described based on the information about the set of singular points.

The necessary theorems are as follows:

Theorem 5

([19]) Let X be a Fano 3-fold with Fano index $f \ge 3$. In this case, the Hilbert series of X with basket of singularities $\{[a_k,~r_k]\}^m_{k=1}$ is given by the following.

$$\begin{aligned} P(X, t):= & {} \sum _{n = 0}^{\infty } P_{n}(X)t^{n}\hspace{1cm}\\= & {} \frac{1}{1-t}+\frac{\bigg (f^{2}+3f+2\bigg )t+\bigg (-2f^{2}+ 8\bigg )t^{2}+\bigg (f^{2}-3f + 2\bigg )t^{3}}{12(1-t)^{4}}A^{3}\\{} & {} + \frac{t}{(1-t)^{2}}\frac{A\cdot c_{2}(X)}{12}\\{} & {} + \sum _{k=1}^{m} \frac{1}{1-t^{r_{k}}}\left( \sum _{l=1}^{r_{k}-1}\left( -i_{k, l}\frac{r_{k}^{2}-1}{12r_{k}}+ \sum _{j=1}^{i_{k, l}-1}\frac{\overline{b_{k}j}\bigg (r_{k}-\overline{b_{k}j}\bigg )}{2r_{k}}\right) t^{l}\right) . \end{aligned}$$

where, $i_{k,l} \in [0,r_k-1]$ represents the local index of nA at each singular point, and $b_k$ is the smallest positive integer satisfying $a_kb_k\equiv 1$.

If $f\ge 3$, we have

Corollary 1

One has

$$\begin{aligned} A^{3}= & {} \frac{12}{(f-1)(f-2)}\hspace{12.5cm}\\{} & {} {}\times \left( 1 - \frac{A\cdot c_{2}(X)}{12} + \sum _{k=1}^{m} \left( -i_{k, -1}\frac{r_{k}^{2} -1}{12r_{k}} + \sum _{j=1}^{i_{k,l}-1} \frac{\overline{b_{k}j}\bigg (r_{k}-\overline{b_{k}j}\bigg )}{2r_{k}}\right) \right) . \end{aligned}$$

For $f=2$, see [4, 17]).

In [19], we used this graded ring method with Hilbert series to calculate the Fano index. At first, we computed all the Basket of singularities satisfying the following three conditions for each Fano index $f \ge 2$. These calculations were done by using the Magma software ( [7, 12]), and the resulting data were shared in [5] together with another results of Fano index 1 [1, 2, 8, 14].

Theorem 6

(1):

$$\begin{aligned} -K^3_X>0 \end{aligned}$$

(2):

(Kawamata-Vieweg vanishing theorem, [9])

$$\begin{aligned} \chi (nA) = \dim H^0(nA) \quad \forall n > -f \end{aligned}$$

(3):

(Kawamata boundedness theorem, [10, 19])

$$\begin{aligned} \bigg (4f^2-3f\bigg )A^3 \le 4 \bigg (-K_X\cdot c_2(X)\bigg ). \end{aligned}$$

Next, we provide a detailed exposition of the Kawamata Boundedness Theorem, which played a significant role in determining the maximum value of the Fano index among the three constraint conditions. We elaborate on the original theorem in [10] and its reconstruction in [19].

2 Kawamata boundedness theorem and Bogomolov stability bound

Theorem 7

(Kawamata boundedness theorem [10])

There exist a universal constant $b>0$ such that all Fano 3-folds X have

$$\begin{aligned} \bigg (-K_X\bigg )^3 \le b\bigg (-K_X\cdot c_2(X)\bigg ). \end{aligned}$$

In particular, $0<(-K_X\cdot c_2(X))$.

Proposition 1

([10], pp. 442–443) Let X be a Fano 3-fold and ${\mathcal {E}}:= (\Omega ^1 _X )^{**}$ the double dual of the sheaf of Kähler differentials of X. If ${\mathcal {E}}$ is not $\mu$-semistable, one can take the maximal destabilizing sheaf ${\mathcal {F}}$ of ${\mathcal {E}}$, which is a (unique) $\mu$-semistable subsheaf ${\mathcal {F}}\subset {\mathcal {E}}$. Then ${\mathcal {F}}$ is necessarily reflexive and of rank $s = 1$ or 2, such that

$$\begin{aligned} \frac{ c_1({\mathcal {F}})\cdot \bigg (-K_X \bigg )^2}{s} > \frac{-K_X ^3}{ 3}. \end{aligned}$$

Remark 1

From the definition of X, ${{\,\textrm{Pic}\,}}X \cong \mathbb Z$ and $\mathbb {Q}$-decomposition leads $c_1({\mathcal {F}}) \equiv tK_X$ for $0<t<s/3$.

Theorem 8

(Kawamata boundedness theorem [20]) Under above assumptions, we have the following:

(1):

If ${\mathcal {E}}$ is $\mu$-semistable, then

$$\begin{aligned} \bigg (-K_X \bigg )^3 \le 3\bigg (-K_X\cdot c_2(X)\bigg ). \end{aligned}$$

(2):

If ${\mathcal {E}}$ is not $\mu$-semistable, and $s = 1$, then one of the following holds:

(a):

$(1 -t)(1 + 3t)(-K_X )^3 \le 4(-K_X \cdot c_2(X)),$ or

(b):

$(tu + (t + u)(1 - t -u))(-K_X )^3 \le (-K_X \cdot c_2(X))$ for some rational number u such that $t< u < 1 - t - u$.

(3):

If ${\mathcal {E}}$ is not $\mu$-semistable and $s = 2$, then

$$\begin{aligned} t(4-3t)\bigg (-K_X^3\bigg ) \le 4\bigg (-K_X\cdot c_2(X)\bigg ). \end{aligned}$$

Remark 2

The inequality (1) is called the Bogomolov stability bound.

Corollary 2

One has

$$\begin{aligned}(4f^2-3f)A^3 \le 4(-K_X\cdot c_2(X)).\end{aligned}$$

3 Main results

For Fano index $f\ge 9$, hand calculations confirm that X satisfies the Bogomolov stability bound. Therefore, we next consider the case of $1\le f\le 8$.

The table below shows the upper bound of positive rational numbers k satisfying $A^3 = \frac{k}{12}A\cdot c_2(X)$ for each index f, with the decimal part truncated to five digits.

For Fano 3-fold X, we have the following theorems.

Theorem 9

(1):

If $f(X)=1$, all X satisfies the Bogomolov stability bound.

(2):

If $f(X)=2, 3$ or 8, at least one of the following holds.

(a):

X satisfies the Bogomolov stability bound.

(b):

${\mathcal {E}}$ is not $\mu$-semistable. Moreover, the maximal subsheaf ${\mathcal {F}}\subset {\mathcal {E}}$ is reflexive and ${{\,\textrm{rank}\,}}{\mathcal {F}}= 2$.

We use the following Lemma.

Lemma 10

([19]) For $c_1({\mathcal {F}})=tK_X$, we have $0<t<s/3$ and $t\in \{1/f,2/f,\dots \}$.

Proof of Theorem 9

(1):

If X does not satisfy the Bogomolov stability bound, from Lemma 10, for $c_1({\mathcal {F}})=tK_X$, in the case of $s=1$ then $0<t <1/3$ and $t \in \{1,2,3,\dots \}$ should occur but there is no such t. This is a contradiction. Similarly, if $s=2$, then $0< t < 2/3$ and $t \in \{1, 2,3,\dots \}$, but there is no such t. Contradiction.

(2):

For $f=2$, if X does not satisfy the Bogomolov stability bound, then from Lemma 10, for $c_1({\mathcal {F}})=tK_X$ and if $s =1$ then we have $0<t<1/3$ and $t \in \{1/2, 1, 3/2,\dots \}$ should hold. Contradicition. Similarly, if $s=2$, $0<t<2/3$ and $t\in \{1/2, 1, 3/2, \dots \}$. This can only happen when $t=1/2$. The proof in the case $f=3$ is similar. For $f=8$, if X does not satisfy the Bogomolov stability bound, then from Lemma 10, for $c_1({\mathcal {F}})=tK_X$ and if $s =1$ then we have $0<t<1/3$ and $t \in \{1/8, 1/4, 3/8,\dots \}$ should hold so $t=1/8$ or 1/4. Similarly, if $s=2$, $0<t<2/3$ and $t\in \{1/8, 1/4, 3/8, \dots \}$. This can only happen when $t=1/8, 1/4, 3/8, 1/2$ or 5/8. The Table 1 shows that only $t=1/8$ case for $s=2$ should hold.

$\square$

Corollary 3

Suppose that a Fano 3-fold X does not satisfy the Bogomolov stability bound, then for the case of $f(X)=2,~3,~8$, we have

$$\begin{aligned} c_1({\mathcal {F}})\equiv 1/f(X)~K_X. \end{aligned}$$

Similarly, for the case of $4 \le f \le 7$ with data of GRDB, we have

Theorem 11

For Fano 3-fold X with $4 \le f(X) \le 7$, at least one of the following holds.

(a):

X satisfies the Bogomolov stability bound.

(b):

${\mathcal {E}}$ is not $\mu$-semistable. Moreover, the maximal subsheaf ${\mathcal {F}}\subset {\mathcal {E}}$ is reflexive and ${{\,\textrm{rank}\,}}{\mathcal {F}}= 2$.

(c):

X is in the Table 2 below.

Table 1 List of upper bound of k

Table 2 List of Fano3-folds

Corollary 4

Suppose that a Fano 3-fold X satisfies the condition (b) of theorem 11 Then we have

$$\begin{aligned} c_1({\mathcal {F}})\equiv 1/f(X)~K_X \end{aligned}$$

4 List of Fano 3-folds that do not satisfy the Bogomolov stability bound

From GRDB, the number of Fano 3-fold, including cases where non-existence has been proven for $f=1$, that do not satisfy the the Bogomolov stability bound is as follows:

$$\begin{aligned} \begin{array}{|c|c|c|c|c|c|c|c|c|c|} \hline f &{} 1 &{} 2 &{} 3 &{} 4 &{} 5&{} 6 &{} 7 &{}8 &{} \ge 9 \\ \hline \#&{} 13096 &{} 79 &{} 51 &{} 34 &{} 19 &{} 3 &{} 4 &{} 2 &{} 0\\ \hline \end{array} \end{aligned}$$

Conjecture 1

All Fano 3-folds with Picard number 1 satisfy the Bogomolov stability bound.

Remark 3

The more general case of Picard number $\ge 2$ is harder.

In addition to the previously obtained results for $f\ge 9$, this paper supports the correctness of the conjecture for $f=1$. Moreover, for $2\le f\le 8$, the following table is a speculative list of 3-dimensional Fano varieties to be removed from GRDB in future. We hope to address this in future research.

Finally, we have the list of numerical possibilities for Fano 3-folds not satisfying the Bogomolov stability bound in the next section.

We omit the case $f=1$, since it involves an extremely large number of cases (more than 10,000), and the list is easily obtained from GRDB by inputting an additional condition.

f	#	Basket	$A^3$	$Ac_2/12$	k
2	39729	2 $\times$ 1/3(1,2,2), 1/7(2,2,5), 1/11(2,5,6)	79/231	26/693	9.11
2	39831	2 $\times$ 1/7(2,3,4), 1/9(2,2,7)	34/63	11/189	9.27
2	39832	1/5(2,2,3), 1/7(2,2,5), 1/11(2,4,7)	211/385	38/65	9.17
2	39843	2 $\times$ 1/5(2,2,3), 1/13(2,3,10)	38/65	4/65	9.50
2	39844	1/3(1,2,2), 1/7(2,3,4), 1/13(2,5,8)	160/273	53/819	9.05
2	39850	4 $\times$ 1/3(1,2,2), 1/5(2,2,3), 1/7(2,3,4)	68/105	22/315	9.27
2	40031	1/5(1,2,4), 1/7(2,3,4), 1/11(1,2,10)	219/385	23/385	9.50
2	40042	1/3(1,2,2), 1/9(1,2,8), 1/11(2,5,6)	59/99	19/297	9.52
2	40046	1/3(1,2,2), 1/7(1,2,6), 1/13(2,6,7)	166/273	53/819	9.39
2	40134	1/9(1,2,8), 1/13(2,3,10)	97/117	32/351	9.09
2	40168	1/3(1,2,2), 1/19(2,5,14)	52/57	17/171	9.17
2	40220	3 $\times$ 1/5(1,2,4), 1/7(2,2,5)	37/35	4/35	9.25
2	40236	1/3(1,2,2), 1/5(2,2,3), 1/7(1,2,6), 1/7(2,3,4)	113/105	37/315	9.16
2	40244	1/3(1,2,2), 1/5(1,2,4), 1/5(2,2,3), 1/9(2,4,5)	10/9	16/135	9.37
2	40263	2 $\times$ 1/3(1,2,2), 1/5(1,2,4), 1/11(2,4,7)	191/165	61/495	9.39
2	40264	2 $\times$ 1/3(1,2,2), 1/7(2,3,4), 1/9(2,4,5)	73/63	23/189	9.50
2	40338	1/5(1,2,4), 1/5(2,2,3), 1/11(2,2,9)	15/11	8/55	9.37
2	40339	1/5(1,2,4), 1/7(2,2,5), 1/9(2,2,7)	431/315	136/945	9.50
2	40348	1/3(1,2,2), 1/7(2,3,4), 1/11(2,2,9)	326/231	103/693	9.49
2	40352	5 $\times$ 1/3(1,2,2), 1/7(2,2,5)	32/21	10/63	9.60
2	40387	1/5(1,2,4), 1/17(1,2,16)	74/85	8/85	9.25
2	40470	1/3(1,2,2), 1/5(2,2,3), 1/13(1,2,12)	272/195	88/585	9.27
2	40520	1/9(2,4,5), 1/11(2,5,6)	158/99	52/297	9.11
2	40566	4 $\times$ 1/3(1,2,2), 1/9(1,2,8)	16/9	5/27	9.60
2	40583	1/3(1,2,2), 2 $\times$ 1/5(1,2,4), 1/7(2,3,4)	194/105	64/315	9.09
2	40608	2 $\times$ 1/3(1,2,2), 1/7(1,2,6), 1/7(2,2,5)	41/21	13/63	9.09
2	40641	1/5(1,2,4), 1/5(2,2,3), 1/9(2,2,7)	19/9	31/135	9.19

f	#	Basket	$A^3$	$Ac_2/12$	k
2	40642	1/5(1,2,4), 2 $\times$ 1/7(2,2,5)	74/35	8/35	9.25
2	40648	1/3(1,2,2), 1/7(2,3,4), 1/9(2,2,7)	136/63	44/189	9.27
2	40649	1/5(2,2,3), 1/7(2,2,5), 1/7(2,3,4)	76/35	8/35	9.50
2	40652	2 $\times$ 1/3(1,2,2), 1/13(2,5,8)	86/39	28/117	9.21
2	40653	1/3(1,2,2), 1/5(2,2,3), 1/11(2,3,8)	364/165	116/495	9.41
2	40654	5 $\times$ 1/3(1,2,2), 1/5(2,2,3)	34/15	11/45	9.27
2	40660	2 $\times$ 1/3(1,2,2), 1/13(2,2,11)	89/39	28/117	9.53
2	40719	1/3(1,2,2), 1/5(1,2,4), 1/11(1,2,10)	361/165	116/495	9.33
2	40723	1/3(1,2,2), 1/7(1,2,6), 1/9(1,2,8)	39/63	44/189	9.47
2	40751	1/5(2,2,3), 1/13(2,6,7)	159/65	17/65	9.35
2	40794	2$\times$ 1/3(1,2,2), 1/5(2,2,3), 1/7(1,2,6)	283/105	92/315	9.22
2	40798	1/3(1,2,2), 2 $\times$ 1/5(1,2,4), 1/5(2,2,3)	41/15	13/45	9.46
2	40805	1/17(2,7,10)	47/17	5/17	9.40
2	40806	3 $\times$ 1/3(1,2,2), 1/9(2,4,5)	25/9	8/27	9.37
2	40807	2 $\times$ 1/3(1,2,2), 1/5(1,2,4), 1/7(2,3,4)	292/105	92/315	9.52
2	40818	1/5(1,2,4), 1/5(2,2,3), 1/7(2,2,5)	20/7	11/35	9.09
2	40823	1/3(1,2,2), 1/7(2,2,5), 1/7(2,3,4)	61/21	20/63	9.15
2	40824	2 $\times$ 1/5(2,2,3), 1/7(2,3,4)	102/35	11/35	9.27
2	40826	2 $\times$ 1/3(1,2,2), 1/11(2,3,8)	97/33	32/99	9.09
2	40832	2 $\times$ 1/3(1,2,2), 1/11(2,2,9)	100/33	32/99	9.37
2	40833	1/3(1,2,2), 1/5(2,2,3), 1/9(2,2,7)	137/45	43/135	9.55
2	40834	1/3(1,2,2), 2 $\times$ 1/7(2,2,5)	64/21	20/63	9.60
2	40877	1/7(2,3,4), 1/9(1,2,8)	199/63	65/189	9.18
2	40881	1/3(1,2,2), 1/13(2,6,7)	124/39	41/117	9.07
2	40884	1/7(1,2,6), 1/9(2,4,5)	202/63	65/189	9.32
2	40886	1/5(1,2,4), 1/11(2,5,6)	177/55	19/55	9.31
2	40892	1/7(2,2,5), 1/9(1,2,8)	208/63	65/189	9.60
2	40909	2 $\times$ 1/3(1,2,2), 2 $\times$ 1/5(1,2,4)	52/15	17/45	9.17
2	40910	1/15(2,4,11)	52/15	17/45	9.17
2	40923	1/3(1,2,2), 1/5(2,2,3), 1/7(2,3,4)	383/105	127/315	9.04
2	40929	2 $\times$ 1/3(1,2,2), 1/9(2,2,7)	34/9	11/27	9.27
2	40930	1/3(1,2,2), 1/5(2,2,3), 1/7(2,2,5)	398/105	127/315	9.40
2	40931	3 $\times$ 1/5(2,2,3)	19/5	2/5	9.50
2	40939	1/15(1,2,14)	52/15	17/45	9.17
2	40958	1/5(2,2,3), 1/9(1,2,8)	182/45	58/135	9.41
2	40967	1/3(1,2,2), 1/11(2,5,6)	137/33	43/99	9.55
2	40979	1/13(2,3,10)	57/13	6/13	9.50
2	40980	3 $\times$ 1/3(1,2,2), 1/5(1,2,4)	22/5	7/15	9.42
2	40984	2 $\times$ 1/3(1,2,2), 1/7(2,2,5)	95/21	31/63	9.19
2	40985	1/3(1,2,2), 2 $\times$ 1/5(2,2,3)	68/15	22/45	9.27
2	40997	1/3(1,2,2), 1/9(1,2,8)	43/9	14/27	9.21
2	41001	1/5(1,2,4), 1/7(1,2,6)	169/35	18/35	9.38
2	41008	1/11(2,4,7)	56/11	6/11	9.33
2	41012	2 $\times$ 1/3(1,2,2), 1/5(2,2,3)	79/15	26/45	9.11
2	41013	4 $\times$ 1/3(1,2,2)	16/3	5/9	9.60
2	41023	1/3(1,2,2), 1/7(1,2,6)	121/21	38/63	9.55
2	41024	1/9(2,4,5)	52/9	7/27	9.17
2	41034	1/7(2,3,4)	47/7	5/7	9,40
2	41035	1/7(2,2,5)	48/7	5/7	9.60
2	41038	1/5(1,2,4)	37/5	4/5	9.25
2	41039	1/5(2,2,3)	38/5	4/5	9.50
2	41041	1/3(1,2,2)	25/3	8/9	9.50

f	#	Basket	$A^3$	$Ac_2/12$	k
3	41072	2 $\times$ 1/2(1,1,1), 1/5(2,3,3), 1/7(3,3,4), 1/8(1,3,7)	47/280	137/3360	4.11
3	41080	1/2(1,1,1), 1/5(1,3,4), 1/5(2,3,3), 1/11(3,4,7)	5/22	73/1320	4.10
3	41089	5 $\times$ 1/2(1,1,1), 2 $\times$ 1/4(1,3,3), 1/7(1,3,6)	2/7	5/84	4.80
3	41093	1/4(1,3,3), 1/7(3,3,4), 1/11(2,3,9)	93/308	85/1232	4.37
3	41094	2 $\times$ 1/2(1,1,1), 1/5(1,3,4), 1/7(2,3,5), 1/7(3,3,4)	11/35	29/420	4.37
3	41096	4 $\times$ 1/2(1,1,1), 1/5(1,3,4), 1/11(3,4,7)	18/55	7/110	5.14
3	41120	1/2(1,1,1), 1/5(2,3,3), 1/16(1,3,15)	17/80	47/960	4.34
3	41129	2 $\times$ 1/2(1,1,1), 1/8(1,3,7), 1/11(1,3,10)	23/88	65/1056	4.24
3	41145	1/5(1,3,4), 1/16(3,5,11)	33/80	29/320	4.55
3	41153	1/2(1,1,1), 1/7(2,3,5), 1/13(1,3,12)	69/182	55/728	5.01
3	41166	1/2(1,1,1), 1/5(2,3,3), 2 $\times$ 1/7(1,3,6)	33/70	31/280	4.25
3	41171	2 $\times$ 1/4(1,3,3), 1/5(2,3,3), 1/8(1,3,7)	21/40	17/160	4.94
3	41172	1/4(1,3,3), 1/5(1,3,4), 1/5(2,3,3), 1/7(1,3,6)	15/28	59/560	5.08
3	41174	1/2(1,1,1), 2 $\times$ 1/4(1,3,3), 1/11(3,5,6)	6/11	5/44	4.80
3	41175	1/2(1,1,1), 1/4(1,3,3), 3 $\times$ 1/5(1,3,4)	11/20	29/240	4.55
3	41178	4 $\times$ 1/2(1,1,1), 1/4(1,3,3), 1/10(1,3,9)	11/20	29/240	4.55
3	41179	4 $\times$ 1/2(1,1,1), 2 $\times$ 1/7(1,3,6)	4/7	5/42	4.80
3	41182	1/2(1,1,1), 1/4(1,3,3), 1/7(1,3,6), 1/7(2,3,5)	17/28	47/336	4.34
3	41187	1/2(1,1,1), 1/5(2,3,3), 1/13(3,4,9)	87/130	69/520	5.04
3	41188	2 $\times$ 1/4(1,3,3), 1/5(1,3,4), 1/7(2,3,5)	47/70	113/840	4.99
3	41192	4 $\times$ 1/2(1,1,1), 1/5(1,3,4), 1/7(3,3,4)	26/35	37/210	4.21
3	41194	2 $\times$ 1/2(1,1,1), 1/4(1,3,3), 1/5(2,3,3), 1/7(3,3,4)	111/140	87/560	5.10
3	41207	1/5(2,3,3), 1/13(1,3,12)	46/65	34/195	4.05
3	41216	2 $\times$ 1/5(1,3,4), 1/8(1,3,7)	33/40	29/160	4.55
3	41217	1/7(1,3,6), 1/11(3,5,6)	64/77	40/231	4.80
3	41221	3 $\times$ 1/2(1,1,1), 1/13(1,3,12)	21/26	19/104	4.42
3	41224	1/7(2,3,5), 1/10(1,3,9)	61/70	169/840	4.42
3	41227	2 $\times$ 1/2(1,1,1), 1/4(1,3,3), 1/11(1,3,10)	39/44	31/176	5.03
3	41229	2 $\times$ 1/2(1,1,1), 1/5(1,3,4), 1/10(1,3,9)	9/10	7/40	5.14
3	41230	2 $\times$ 1/2(1,1,1), 1/7(1,3,6), 1/8(1,3,7)	51/56	39/224	5.23
3	41235	3 $\times$ 1/2(1,1,1), 1/4(1,3,3), 1/7(1,3,6)	29/28	83/336	4.19
3	41238	2 $\times$ 1/2(1,1,1), 2 $\times$ 1/4(1,3,3), 1/5(1,3,4)	11/10	29/120	4.55
3	41239	1/5(2,3,3), 1/11(3,4,7)	62/55	38/165	4.89
3	41240	3 $\times$ 1/4(1,3,3), 1/5(2,3,3)	23/20	53/240	5.20
3	41242	1/2(1,1,1), 1/7(2,3,5), 1/7(3,3,4)	17/14	41/168	4.97
3	41243	3 $\times$ 1/2(1,1,1), 1/11(3,4,7)	27/22	21/88	5.14
3	41248	1/2(1,1,1), 1/14(1,3,13)	8/7	5/21	4.80
3	41253	2 $\times$ 1/2(1,1,1), 1/10(1,3,9)	13/10	37/120	4.21
3	41255	1/2(1,1,1), 1/4(1,3,3), 1/8(1,3,7)	11/8	29/96	4.55
3	41256	1/2(1,1,1), 1/5(1,3,4), 1/7(1,3,6)	97/70	253/840	4.60
3	41257	1/4(1,3,3), 2 $\times$ 1/5(1,3,4)	29/20	71/240	4.90
3	41259	2 $\times$ 1/2(1,1,1), 1/4(1,3,3), 1/7(1,3,6)	43/28	97/336	5.31
4	41282	2 $\times$ 1/3(1,1,2), 1/5(2,3,4), 1/13(4,6,7)	11/195	23/1170	2.86
4	41286	2 $\times$ 1/5(1,4,4), 1/13(3,4,10)	6/65	2/65	3.00
4	41294	2 $\times$ 1/7(2,4,5), 1/9(1,4,8)	5/63	11/378	3.00
4	41296	1/5(1,4,4), 1/7(1,4,6), 1/11(2,4,9)	37/385	23/770	3.21
4	41303	2 $\times$ 1/3(1,1,2), 2 $\times$ 1/5(1,4,4), 1/7(2,4,5)	17/105	29/630	3.51
4	41309	1/3(1,1,2), 1/5(1,4,4), 1/13(2,4,11)	34/195	44/585	2.31
4	41310	1/5(1,4,4), 1/5(2,3,4), 1/11(2,4,9)	2/11	4/55	2.50
4	41311	1/5(1,4,4), 1/7(2,4,5), 1/9(2,4,7)	58/315	68/945	2.55
4	41315	1/3(1,1,2), 1/5(2,3,4), 1/13(4,5,8)	46/195	44/585	3.13
4	41329	1/7(2,4,5), 1/15(1,4,14)	17/105	29/630	3.51

f	#	Basket	$A^3$	$Ac_2/12$	k
4	41336	3 $\times$ 1/3(1,1,2), 1/13(1,4,12)	3/13	5/78	3.60
4	41345	1/9(2,4,7), 1/11(1,4,10)	28/99	26/297	3.23
4	41348	1/3(1,1,2), 1/7(2,4,5), 1/9(1,4,8)	20/63	22/189	2.72
4	41350	1/7(2,4,5), 1/11(4,5,6)	26/77	10/77	2.60
4	41351	1/5(1,4,4), 1/7(1,4,6), 1/7(2,4,5)	12/35	4/35	3.00
4	41353	2 $\times$ 1/3(1,1,2), 1/5(2,3,4), 1/9(1,4,8)	16/45	14/135	3.42
4	41354	1/3(1,1,2), 1/5(2,3,4), 1/11(4,5,6)	62/165	58/495	3.20
4	41355	3 $\times$ 1/3(1,1,2), 2 $\times$ 1/5(1,4,4)	2/5	2/15	3.00
4	41356	1/3(1,1,2), 1/7(3,4,4), 1/9(4,4,5)	26/63	22/189	3.54
4	41361	1/3(1,1,2), 1/5(1,4,4), 1/9(2,4,7)	19/45	43/270	2.65
4	41362	1/5(1,4,4), 1/5(2,3,4), 1/7(2,4,5)	3/7	11/70	2.72
4	41363	1/3(1,1,2), 1/13(4,5,8)	17/39	41/234	2.48
4	41364	1/3(1,1,2), 1/5(1,4,4), 2 $\times$ 1/5(2,3,4)	7/15	13/90	3.23
4	41365	1/17(4,5,12)	9/17	5/34	3.60
4	41369	1/3(1,1,2), 1/15(1,4,14)	2/5	2/15	3.00
4	41371	1/7(1,4,6), 1/11(1,4,10)	34/77	10/77	3.60
4	41375	1/5(2,3,4), 1/11(1,4,10)	29/55	19/110	3.60
4	41377	2 $\times$ 1/3(1,1,2), 1/9(1,4,8)	5/9	11/54	2.72
4	41378	1/3(1,1,2), 1/11(4,5,6)	19/33	43/198	2.65
4	41379	1/3(1,1,2), 1/5(1,4,4), 1/7(1,4,6)	61/105	127/630	2.88
4	41380	1/3(1,1,2), 1/13(4,6,7)	23/39	41/234	3.36
4	41383	1/5(1,4,4), 1/7(2,4,5)	22/35	9/35	2.44
5	41402	1/2(1,1,1), 1/4(1,1,3), 1/6(1,5,5), 1/11(4,5,7)	7/132	53/1584	1.58
5	41403	1/3(1,2,2), 1/6(1,5,5), 2 $\times$ 1/7(3,4,5)	1/14	5/168	2.40
5	41408	1/7(1,5,6), 1/7(2,5,5), 1/8(3,5,5)	5/56	9/224	2.22
5	41410	2 $\times$ 1/2(1,1,1), 2 $\times$ 1/3(1,2,2), 1/6(1,5,5), 1/7(3,4,5)	5/42	25/504	2.40
5	41411	1 4 $\times$ 1/2(1,1,1), 3 $\times$ 1/3(1,2,2), 1/6(1,5,5)	1/6	5/72	2.40
5	41421	1/4(1,1,3), 1/17(5,8,9)	7/68	15/272	1.86
5	41424	2 $\times$ 1/2(1,1,1), 1/3(1,2,2), 1/14(1,5,13)	5/42	37/504	1.62
5	41426	2 $\times$ 1/2(1,1,1), 1/4(1,1,3), 1/13(1,5,12)	7/52	15/208	1.86
5	41427	2 $\times$ 1/2(1,1,1), 1/8(1,5,7), 1/9(1,5,8)	11/72	61/864	2.36
5	41431	1/2(1,1,1), 1/3(1,2,2), 1/4(1,1,3), 1/12(1,5,11)	1/6	5/72	2.40
5	41433	1/2(1,1,1), 1/3(1,2,2), 1/7(1,5,6), 1/9(1,5,8)	23/126	103/1512	2.67
5	41435	1/2(1,1,1), 2 $\times$ 1/4(1,1,3), 1/11(1,5,10)	2/11	3/44	2.66
5	41436	1/3(1,2,2), 1/7(1,5,6), 1/7(3,4,5)	4/21	8/63	1.50
5	41437	1/4(1,1,3), 1/6(1,5,5), 1/7(3,4,5)	17/84	127/1008	1.60
5	41438	1/7(1,5,6), 1/11(4,5,7)	16/77	8/77	2.00
5	41441	2 $\times$ 1/2(1,1,1), 2 $\times$ 1/3(1,2,2), 1/7(1,5,6)	5/21	37/252	1.62
5	41442	2 $\times$ 1/2(1,1,1), 1/3(1,2,2), 1/4(1,1,3), 1/6(1,5,5)	1/4	7/48	1.71
5	41443	1/2(1,1,1), 1/3(1,2,2), 1/6(1,5,5), 1/7(3,4,5)	2/7	5/42	2.40
5	41447	1/2(1,1,1), 1/14(1,5,13)	2/7	1/7	2.00
6	41463	1/7(3,4,6), 1/13(2,6,11)	10/91	16/273	1.87
6	41467	1/7(2,5,6), 1/11(1,6,10)	8/77	20/231	1.20
6	41468	1/5(1,1,4), 1/5(1,2,3), 1/7(1,6,6)	1/7	11/105	1.36
7	41473	4 $\times$ 1/2(1,1,1), 1/5(2,2,3), 1/12(1,7,11)	1/60	11/720	1.09
7	41482	2 $\times$ 1/2(1,1,1), 1/3(1,1,2), 1/12(1,7,11)	1/12	11/144	1.09
7	41483	2 $\times$ 1/2(1,1,1), 1/5(1,2,4), 1/10(1,7,9)	1/10	3/40	1.33
7	41485	2 $\times$ 1/3(1,1,2), 1/11(1,7,10)	7/66	59/792	1.42
8	41496	1/2(1,1,1), 2 $\times$ 1/3(1,2,2), 1/5(1,3,4), 1/9(2,7,8)	2/45	7/135	1.16
8	41498	1/7(1,3,4), 1/13(1,8,12)	4/9	4/91	1.00