Calabi-Yau differential operator database v.3.0

1

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

Degree: 2

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

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

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

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

Note:

This is operator "2.64" from ...

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

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

2

New Number: 3.28 | AESZ: 410 | Superseeker: 7 1057/3 | Hash: accbbff67291992dfbc89e78f5a3c897

Degree: 3

$\theta^4-x\left(145\theta^4+242\theta^3+199\theta^2+78\theta+12\right)+2^{3} x^{2}(2\theta+1)(4\theta+3)(97\theta^2+182\theta+114)-2^{4} 3^{4} x^{3}(2\theta+1)(2\theta+3)(4\theta+3)(4\theta+7)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 336, 12880, 592200, ...
--> OEIS
Normalized instanton numbers (n₀=1): 7, 22, 1057/3, 5460, 108241, ... ; Common denominator:...

Discriminant

$-(81z-1)(-1+32z)^2$

Local exponents

$0$ $\frac{ 1}{ 81}$ $\frac{ 1}{ 32}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $\frac{ 3}{ 4}$ $\frac{ 3}{ 4}$
$0$ $1$ $1$ $\frac{ 3}{ 2}$
$0$ $2$ $\frac{ 7}{ 4}$ $\frac{ 7}{ 4}$

\(0\)	\(\frac{ 1}{ 81}\)	\(\frac{ 1}{ 32}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(\frac{ 3}{ 4}\)	\(\frac{ 3}{ 4}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 3}{ 2}\)
\(0\)	\(2\)	\(\frac{ 7}{ 4}\)	\(\frac{ 7}{ 4}\)

Note:

This is operator "3.28" from ...

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

3

New Number: 10.8 | AESZ: | Superseeker: 7 -2044/9 | Hash: 772d055ae4c1a5d6a65a2b1f3ffa351b

Degree: 10

$\theta^4-x\left(147\theta^2+10+60\theta+174\theta^3+111\theta^4\right)+2^{2} x^{2}\left(1269\theta^4+3576\theta^3+4595\theta^2+2722\theta+639\right)-2^{2} x^{3}\left(28236\theta^4+92256\theta^3+135641\theta^2+100407\theta+29996\right)+2^{4} 3 x^{4}\left(34932\theta^4+117280\theta^3+166025\theta^2+128238\theta+41467\right)-2^{6} x^{5}\left(266139\theta^4+937698\theta^3+1398643\theta^2+1056533\theta+325061\right)+2^{8} x^{6}\left(478785\theta^4+1758504\theta^3+2952901\theta^2+2388960\theta+754208\right)-2^{8} x^{7}\left(2371176\theta^4+9770640\theta^3+17775969\theta^2+15468753\theta+5209610\right)+2^{10} x^{8}\left(1853604\theta^4+9368112\theta^3+18957629\theta^2+17669710\theta+6248237\right)-2^{12} 11 x^{9}(2\theta+3)(36502\theta^3+178659\theta^2+286703\theta+145866)+2^{16} 3 5^{2} 11^{2} x^{10}(\theta+1)(2\theta+5)(2\theta+3)(\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 10, 154, 2548, 27370, ...
--> OEIS
Normalized instanton numbers (n₀=1): 7, -31/4, -2044/9, -1380, -8520, ... ; Common denominator:...

Discriminant

$(3z-1)(6400z^3-2352z^2+84z-1)(4z-1)^2(88z^2-8z+1)^2$

Local exponents

$0$ ≈$0.019222-0.010265I$ ≈$0.019222+0.010265I$ $\frac{ 1}{ 22}-\frac{ 3}{ 44}\sqrt{ 2}I$ $\frac{ 1}{ 22}+\frac{ 3}{ 44}\sqrt{ 2}I$ $\frac{ 1}{ 4}$ ≈$0.329056$ $\frac{ 1}{ 3}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$0$ $1$ $1$ $1$ $1$ $\frac{ 1}{ 2}$ $1$ $1$ $\frac{ 3}{ 2}$
$0$ $1$ $1$ $3$ $3$ $\frac{ 1}{ 2}$ $1$ $1$ $\frac{ 5}{ 2}$
$0$ $2$ $2$ $4$ $4$ $1$ $2$ $2$ $3$

\(0\)	≈\(0.019222-0.010265I\)	≈\(0.019222+0.010265I\)	\(\frac{ 1}{ 22}-\frac{ 3}{ 44}\sqrt{ 2}I\)	\(\frac{ 1}{ 22}+\frac{ 3}{ 44}\sqrt{ 2}I\)	\(\frac{ 1}{ 4}\)	≈\(0.329056\)	\(\frac{ 1}{ 3}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)	\(1\)	\(1\)	\(\frac{ 3}{ 2}\)
\(0\)	\(1\)	\(1\)	\(3\)	\(3\)	\(\frac{ 1}{ 2}\)	\(1\)	\(1\)	\(\frac{ 5}{ 2}\)
\(0\)	\(2\)	\(2\)	\(4\)	\(4\)	\(1\)	\(2\)	\(2\)	\(3\)

Note:

This is operator "10.8" from ...

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

4

New Number: 13.11 | AESZ: | Superseeker: 7 -2044/9 | Hash: d6e183df7853fe5068c8b8cdeb3f63cb

Degree: 13

\(\theta^4-x\left(98\theta^4+164\theta^3+137\theta^2+55\theta+9\right)+x^{2}\left(3822\theta^4+11400\theta^3+14901\theta^2+8746\theta+2007\right)-x^{3}\left(64148\theta^4+196344\theta^3+271665\theta^2+199855\theta+60354\right)+x^{4}\left(802771\theta^4+2242504\theta^3+2203855\theta^2+1316868\theta+390636\right)-2 3 x^{5}\left(1040145\theta^4+2982426\theta^3+3578912\theta^2+1897395\theta+345411\right)+2 3^{2} x^{6}\left(1927994\theta^4+4917832\theta^3+7329041\theta^2+5154630\theta+1338003\right)-2 3^{5} x^{7}\left(219316\theta^4+761432\theta^3+1064075\theta^2+703129\theta+181966\right)+3^{4} x^{8}\left(754759\theta^4+7471824\theta^3+13904030\theta^2+8830464\theta+1544112\right)+3^{7} x^{9}\left(174966\theta^4+736236\theta^3+1307237\theta^2+1340471\theta+568265\right)-3^{10} x^{10}(\theta+1)(8018\theta^3+62342\theta^2+139257\theta+108861)-3^{9} x^{11}(\theta+1)(\theta+2)(28988\theta^2+81396\theta+36331)+3^{12} x^{12}(\theta+3)(\theta+2)(\theta+1)(1061\theta+5386)+2 3^{15} 17 x^{13}(\theta+1)(\theta+2)(\theta+3)(\theta+4)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 9, 135, 2115, 18063, ...
--> OEIS
Normalized instanton numbers (n₀=1): 7, -31/4, -2044/9, -1380, -8520, ... ; Common denominator:...

Discriminant

$(2z-1)(4131z^3-2187z^2+81z-1)(3z-1)^2(81z^2-6z+1)^2(z+1)^3$

Local exponents

$-1$ $0$ ≈$0.019487-0.01067I$ ≈$0.019487+0.01067I$ $\frac{ 1}{ 27}-\frac{ 2}{ 27}\sqrt{ 2}I$ $\frac{ 1}{ 27}+\frac{ 2}{ 27}\sqrt{ 2}I$ $\frac{ 1}{ 3}$ ≈$0.490438$ $\frac{ 1}{ 2}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$\frac{ 1}{ 2}$ $0$ $1$ $1$ $1$ $1$ $\frac{ 1}{ 2}$ $1$ $1$ $2$
$\frac{ 3}{ 2}$ $0$ $1$ $1$ $3$ $3$ $\frac{ 1}{ 2}$ $1$ $1$ $3$
$2$ $0$ $2$ $2$ $4$ $4$ $1$ $2$ $2$ $4$

\(-1\)	\(0\)	≈\(0.019487-0.01067I\)	≈\(0.019487+0.01067I\)	\(\frac{ 1}{ 27}-\frac{ 2}{ 27}\sqrt{ 2}I\)	\(\frac{ 1}{ 27}+\frac{ 2}{ 27}\sqrt{ 2}I\)	\(\frac{ 1}{ 3}\)	≈\(0.490438\)	\(\frac{ 1}{ 2}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(\frac{ 1}{ 2}\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)	\(1\)	\(1\)	\(2\)
\(\frac{ 3}{ 2}\)	\(0\)	\(1\)	\(1\)	\(3\)	\(3\)	\(\frac{ 1}{ 2}\)	\(1\)	\(1\)	\(3\)
\(2\)	\(0\)	\(2\)	\(2\)	\(4\)	\(4\)	\(1\)	\(2\)	\(2\)	\(4\)

Note:

This is operator "13.11" from ...

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

5

New Number: 8.2 | AESZ: 104 | Superseeker: 7 1271/3 | Hash: d6bd0d1524954c8ce0a6421d295e9795

Degree: 8

$\theta^4-x(10\theta^2+10\theta+3)(7\theta^2+7\theta+2)-x^{2}\left(71\theta^4+1148\theta^3+1591\theta^2+886\theta+192\right)-2^{3} 3^{2} x^{3}\left(70\theta^4-420\theta^3-1289\theta^2-963\theta-240\right)-2^{4} 3^{2} x^{4}\left(143\theta^4+286\theta^3-1138\theta^2-1281\theta-414\right)+2^{6} 3^{4} x^{5}\left(70\theta^4+700\theta^3+391\theta^2-75\theta-76\right)-2^{6} 3^{4} x^{6}\left(71\theta^4-864\theta^3-1427\theta^2-864\theta-180\right)+2^{9} 3^{6} x^{7}(10\theta^2+10\theta+3)(7\theta^2+7\theta+2)+2^{12} 3^{8} x^{8}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 6, 150, 5208, 221094, ...
--> OEIS
Normalized instanton numbers (n₀=1): 7, 93/2, 1271/3, 18507/2, 190710, ... ; Common denominator:...

Discriminant

$(9z+1)(8z-1)(72z-1)(z+1)(1+72z^2)^2$

Local exponents

$-1$ $-\frac{ 1}{ 9}$ $0-\frac{ 1}{ 12}\sqrt{ 2}I$ $0$ $0+\frac{ 1}{ 12}\sqrt{ 2}I$ $\frac{ 1}{ 72}$ $\frac{ 1}{ 8}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $0$ $1$ $1$ $1$ $1$
$1$ $1$ $3$ $0$ $3$ $1$ $1$ $1$
$2$ $2$ $4$ $0$ $4$ $2$ $2$ $1$

\(-1\)	\(-\frac{ 1}{ 9}\)	\(0-\frac{ 1}{ 12}\sqrt{ 2}I\)	\(0\)	\(0+\frac{ 1}{ 12}\sqrt{ 2}I\)	\(\frac{ 1}{ 72}\)	\(\frac{ 1}{ 8}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(3\)	\(0\)	\(3\)	\(1\)	\(1\)	\(1\)
\(2\)	\(2\)	\(4\)	\(0\)	\(4\)	\(2\)	\(2\)	\(1\)

Note:

Hadamard product $a \ast c$. This operator has a second MUM-point at infinity with the same instanton point.
It is reducible to an operator of degree 4 with a single MUM-point defined over $Q(\sqrt{-2})$

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

Calabi-Yau differential operator database v.3

Summary

Discriminant

Local exponents

Note:

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

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

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

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