Calabi-Yau differential operator database v.3.0

1

New Number: 2.41 | AESZ: | Superseeker: -522 -9879192 | Hash: 73d9f98b2c49f1c35df531f020cf1721

Degree: 2

$\theta^4-2 3^{2} x\left(324\theta^4+648\theta^3+765\theta^2+441\theta+97\right)+2^{2} 3^{10} x^{2}(\theta+1)^2(6\theta+5)(6\theta+7)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 1746, 3951990, 9740271348, 25043989159350, ...
--> OEIS
Normalized instanton numbers (n₀=1): -522, -105291/2, -9879192, -2420127936, -689420749716, ... ; Common denominator:...

Discriminant

$(2916z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 2916}$ $\infty$
$0$ $0$ $\frac{ 5}{ 6}$
$0$ $-\frac{ 1}{ 2}$ $1$
$0$ $1$ $1$
$0$ $\frac{ 3}{ 2}$ $\frac{ 7}{ 6}$

\(0\)	\(\frac{ 1}{ 2916}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 5}{ 6}\)
\(0\)	\(-\frac{ 1}{ 2}\)	\(1\)
\(0\)	\(1\)	\(1\)
\(0\)	\(\frac{ 3}{ 2}\)	\(\frac{ 7}{ 6}\)

Note:

Operator equivalent to $\hat{4}$

	Gopakumar-Vafa invariants
g=0	,...
g=1	,...
g=2	,...

2

New Number: 2.64 | AESZ: 182 | Superseeker: 1 7 | Hash: 89ba4729efa82413b33fe6928ff8d2c9

Degree: 2

$\theta^4-x\left(43\theta^4+86\theta^3+77\theta^2+34\theta+6\right)+2^{2} 3 x^{2}(\theta+1)^2(6\theta+5)(6\theta+7)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 6, 66, 924, 14850, ...
--> OEIS
Normalized instanton numbers (n₀=1): 1, 7/4, 7, 40, 270, ... ; Common denominator:...

Discriminant

$(27z-1)(16z-1)$

Local exponents

$0$ $\frac{ 1}{ 27}$ $\infty$ $\frac{ 1}{ 16}$
$0$ $0$ $\frac{ 5}{ 6}$ $0$
$0$ $1$ $1$ $1$
$0$ $1$ $1$ $1$
$0$ $2$ $\frac{ 7}{ 6}$ $2$

\(0\)	\(\frac{ 1}{ 27}\)	\(\infty\)	\(\frac{ 1}{ 16}\)
\(0\)	\(0\)	\(\frac{ 5}{ 6}\)	\(0\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(2\)	\(\frac{ 7}{ 6}\)	\(2\)

Note:

This is operator "2.64" from ...

	Gopakumar-Vafa invariants
g=0	,...
g=1	,...
g=2	,...

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-96\)	\(132\)	\(132\)	\(33\)

3

New Number: 5.111 | AESZ: 380 | Superseeker: 12 2320 | Hash: 85214e3836a67470a05358a4d38fb124

Degree: 5

$\theta^4-2 x\left(60\theta^4+90\theta^3+68\theta^2+23\theta+3\right)+2^{2} x^{2}\left(313\theta^4-398\theta^3-1417\theta^2-1033\theta-252\right)+2^{3} x^{3}\left(654\theta^4+5064\theta^3+3574\theta^2+129\theta-405\right)-2^{4} 5 x^{4}\left(628\theta^4-40\theta^3-1699\theta^2-1661\theta-480\right)-2^{6} 3 5^{2} x^{5}(\theta+1)^2(6\theta+5)(6\theta+7)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 6, 246, 13020, 832950, ...
--> OEIS
Normalized instanton numbers (n₀=1): 12, 511/4, 2320, 63507, 2180312, ... ; Common denominator:...

Discriminant

$-(108z-1)(4z+1)^2(10z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 108}$ $\frac{ 1}{ 10}$ $\infty$ $-\frac{ 1}{ 4}$
$0$ $0$ $0$ $\frac{ 5}{ 6}$ $0$
$0$ $1$ $1$ $1$ $\frac{ 1}{ 2}$
$0$ $1$ $3$ $1$ $\frac{ 1}{ 2}$
$0$ $2$ $4$ $\frac{ 7}{ 6}$ $1$

\(0\)	\(\frac{ 1}{ 108}\)	\(\frac{ 1}{ 10}\)	\(\infty\)	\(-\frac{ 1}{ 4}\)
\(0\)	\(0\)	\(0\)	\(\frac{ 5}{ 6}\)	\(0\)
\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(3\)	\(1\)	\(\frac{ 1}{ 2}\)
\(0\)	\(2\)	\(4\)	\(\frac{ 7}{ 6}\)	\(1\)

Note:

This is operator "5.111" from ...

	Gopakumar-Vafa invariants
g=0	,...
g=1	,...
g=2	,...

4

New Number: 5.81 | AESZ: 312 | Superseeker: 5/7 48/7 | Hash: 767262575f8b1458839c1e9a8beacf0a

Degree: 5

$7^{2} \theta^4-7 x\left(39\theta^4+234\theta^3+201\theta^2+84\theta+14\right)-2 x^{2}\left(12073\theta^4+43222\theta^3+57461\theta^2+34328\theta+7756\right)-2^{2} x^{3}\left(28923\theta^4+48426\theta^3-33393\theta^2-80976\theta-32032\right)+2^{3} 13 x^{4}\left(359\theta^4+9790\theta^3+20805\theta^2+15784\theta+4124\right)+2^{5} 3 13^{2} x^{5}(\theta+1)^2(6\theta+5)(6\theta+7)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 2, 30, 308, 5950, ...
--> OEIS
Normalized instanton numbers (n₀=1): 5/7, 239/28, 48/7, 4451/14, 5888/7, ... ; Common denominator:...

Discriminant

$(16z+1)(2z-1)(27z-1)(7+26z)^2$

Local exponents

$0$ $\frac{ 1}{ 27}$ $\infty$ $\frac{ 1}{ 2}$ $-\frac{ 1}{ 16}$ $-\frac{ 7}{ 26}$
$0$ $0$ $\frac{ 5}{ 6}$ $0$ $0$ $0$
$0$ $1$ $1$ $1$ $1$ $1$
$0$ $1$ $1$ $1$ $1$ $3$
$0$ $2$ $\frac{ 7}{ 6}$ $2$ $2$ $4$

\(0\)	\(\frac{ 1}{ 27}\)	\(\infty\)	\(\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 16}\)	\(-\frac{ 7}{ 26}\)
\(0\)	\(0\)	\(\frac{ 5}{ 6}\)	\(0\)	\(0\)	\(0\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(3\)
\(0\)	\(2\)	\(\frac{ 7}{ 6}\)	\(2\)	\(2\)	\(4\)

Note:

This is operator "5.81" from ...

	Gopakumar-Vafa invariants
g=0	,...
g=1	,...
g=2	,...

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-108\)	\(108\)	\(84\)	\(23\)

5

New Number: 7.12 | AESZ: | Superseeker: -21 -7941 | Hash: 0841b278bc566a089b643bbe2460fe8b

Degree: 7

$\theta^4+3 x\left(99\theta^4+162\theta^3+139\theta^2+58\theta+10\right)+2 3^{4} x^{2}\left(135\theta^4+738\theta^3+945\theta^2+518\theta+116\right)-2^{2} 3^{7} x^{3}\left(117\theta^4-738\theta^3-2010\theta^2-1493\theta-406\right)-2^{3} 3^{10} x^{4}\left(333\theta^4+774\theta^3-898\theta^2-1269\theta-454\right)-2^{4} 3^{13} x^{5}\left(54\theta^4+1224\theta^3+1179\theta^2+347\theta-22\right)+2^{5} 3^{16} x^{6}\left(180\theta^4+72\theta^3-327\theta^2-359\theta-106\right)+2^{7} 3^{19} x^{7}(\theta+1)^2(6\theta+5)(6\theta+7)$

Maple LaTex

Coefficients of the holomorphic solution: 1, -30, 1458, -89076, 6250050, ...
--> OEIS
Normalized instanton numbers (n₀=1): -21, -399, -7941, -986355/4, -8179455, ... ; Common denominator:...

Discriminant

$(27z+1)(54z+1)(54z-1)^2(108z+1)^3$

Local exponents

$0$ $\frac{ 1}{ 54}$ $\infty$ $-\frac{ 1}{ 54}$ $-\frac{ 1}{ 108}$ $-\frac{ 1}{ 27}$
$0$ $0$ $\frac{ 5}{ 6}$ $0$ $0$ $0$
$0$ $1$ $1$ $1$ $\frac{ 1}{ 2}$ $1$
$0$ $3$ $1$ $1$ $\frac{ 3}{ 2}$ $1$
$0$ $4$ $\frac{ 7}{ 6}$ $2$ $2$ $2$

\(0\)	\(\frac{ 1}{ 54}\)	\(\infty\)	\(-\frac{ 1}{ 54}\)	\(-\frac{ 1}{ 108}\)	\(-\frac{ 1}{ 27}\)
\(0\)	\(0\)	\(\frac{ 5}{ 6}\)	\(0\)	\(0\)	\(0\)
\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)	\(1\)
\(0\)	\(3\)	\(1\)	\(1\)	\(\frac{ 3}{ 2}\)	\(1\)
\(0\)	\(4\)	\(\frac{ 7}{ 6}\)	\(2\)	\(2\)	\(2\)

Note:

This is operator "7.12" from ...

	Gopakumar-Vafa invariants
g=0	,...
g=1	,...
g=2	,...

6

New Number: 8.71 | AESZ: | Superseeker: -15 14044/3 | Hash: de469dbb89801caa07ec523e3b0e4772

Degree: 8

$\theta^4+3 x\left(111\theta^4+186\theta^3+169\theta^2+76\theta+14\right)+2 3^{2} x^{2}\left(2529\theta^4+6930\theta^3+9483\theta^2+6096\theta+1508\right)+2^{2} 3^{4} x^{3}\left(11367\theta^4+32886\theta^3+47658\theta^2+36099\theta+10084\right)+2^{3} 3^{6} x^{4}\left(37017\theta^4+100278\theta^3+103626\theta^2+56025\theta+11582\right)+2^{4} 3^{9} x^{5}\left(29160\theta^4+80676\theta^3+84897\theta^2+27261\theta-568\right)+2^{5} 3^{12} x^{6}\left(16200\theta^4+40824\theta^3+53991\theta^2+31131\theta+6578\right)+2^{7} 3^{17} x^{7}\left(360\theta^4+936\theta^3+1056\theta^2+585\theta+131\right)+2^{9} 3^{20} x^{8}(\theta+1)^2(6\theta+5)(6\theta+7)$

Maple LaTex

Coefficients of the holomorphic solution: 1, -42, 2682, -200436, 16310250, ...
--> OEIS
Normalized instanton numbers (n₀=1): -15, 39, 14044/3, 213069/2, 462576, ... ; Common denominator:...

Discriminant

$(27z+1)(54z+1)(108z+1)^2(1944z^2+18z+1)^2$

Local exponents

$0$ $-\frac{ 1}{ 216}-\frac{ 1}{ 216}\sqrt{ 23}I$ $\infty$ $-\frac{ 1}{ 54}$ $-\frac{ 1}{ 108}$ $-\frac{ 1}{ 216}+\frac{ 1}{ 216}\sqrt{ 23}I$ $-\frac{ 1}{ 27}$
$0$ $0$ $\frac{ 5}{ 6}$ $0$ $0$ $0$ $0$
$0$ $1$ $1$ $1$ $\frac{ 1}{ 6}$ $1$ $1$
$0$ $3$ $1$ $1$ $\frac{ 5}{ 6}$ $3$ $1$
$0$ $4$ $\frac{ 7}{ 6}$ $2$ $1$ $4$ $2$

\(0\)	\(-\frac{ 1}{ 216}-\frac{ 1}{ 216}\sqrt{ 23}I\)	\(\infty\)	\(-\frac{ 1}{ 54}\)	\(-\frac{ 1}{ 108}\)	\(-\frac{ 1}{ 216}+\frac{ 1}{ 216}\sqrt{ 23}I\)	\(-\frac{ 1}{ 27}\)
\(0\)	\(0\)	\(\frac{ 5}{ 6}\)	\(0\)	\(0\)	\(0\)	\(0\)
\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 6}\)	\(1\)	\(1\)
\(0\)	\(3\)	\(1\)	\(1\)	\(\frac{ 5}{ 6}\)	\(3\)	\(1\)
\(0\)	\(4\)	\(\frac{ 7}{ 6}\)	\(2\)	\(1\)	\(4\)	\(2\)

Note:

This is operator "8.71" from ...

\(cy[DM_basis]\)	\(\)	\(\)	\(\)
\(\)	\(\)	\(\)	\(\)
\(\)	\(\)	\(\)	\(\)
\(\)	\(\)	\(\)	\(\)

\(\frac{ 17}{ 4}+288\lambda\)	\(-\frac{ 13}{ 24}-48\lambda\)	\(\frac{ 13}{ 96}+12\lambda\)	\(-.21754163e-1-\frac{ 52}{ 11}\lambda\)
\(\frac{ 33}{ 2}\)	\(-\frac{ 7}{ 4}\)	\(\frac{ 11}{ 16}\)	\(-\frac{ 13}{ 96}-12\lambda\)
\(66\)	\(-11\)	\(\frac{ 15}{ 4}\)	\(-\frac{ 13}{ 24}-48\lambda\)
\(396\)	\(-66\)	\(\frac{ 33}{ 2}\)	\(-\frac{ 9}{ 4}-288\lambda\)

Calabi-Yau differential operator database v.3

Summary

Discriminant

Local exponents

Note:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

\(1\)	\(-1\)	\(\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 6}\)
\(0\)	\(1\)	\(-1\)	\(\frac{ 1}{ 2}\)
\(0\)	\(0\)	\(1\)	\(-1\)
\(0\)	\(0\)	\(0\)	\(1\)

\(\frac{ 7}{ 8}\)	\(-108\lambda\)	\(\frac{ 5}{ 64}\)	\(0\)
\(0\)	\(-\frac{ 7}{ 8}\)	\(0\)	\(\frac{ 5}{ 64}\)
\(3\)	\(0\)	\(-\frac{ 7}{ 8}\)	\(108\lambda\)
\(0\)	\(3\)	\(0\)	\(\frac{ 7}{ 8}\)

\(1+96\lambda\)	\(0\)	\(4\lambda\)	\(.1639606e-2\)
\(\frac{ 11}{ 2}\)	\(1\)	\(\frac{ 11}{ 48}\)	\(-4\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(132\)	\(0\)	\(\frac{ 11}{ 2}\)	\(1-96\lambda\)

\(-96\lambda\)	\(\frac{ 55}{ 2}\)	\(1\)	\(1\)
\(-\frac{ 11}{ 2}\)	\(-66\)	\(-1\)	\(0\)
\(0\)	\(132\)	\(0\)	\(0\)
\(-132\)	\(0\)	\(0\)	\(0\)