Calabi-Yau differential operator database v.3.0

1

New Number: 2.52 | AESZ: 16 | Superseeker: 4 644/3 | Hash: 05af0662662bfbec63e3186c4f363313

Degree: 2

$\theta^4-2^{2} x(2\theta+1)^2(5\theta^2+5\theta+2)+2^{8} x^{2}(2\theta+1)(\theta+1)^2(2\theta+3)$

Coefficients of the holomorphic solution: 1, 8, 168, 5120, 190120, ...
--> OEIS
Normalized instanton numbers (n₀=1): 4, 20, 644/3, 3072, 52512, ... ; Common denominator:...

Discriminant

$(64z-1)(16z-1)$

Local exponents

$0$ $\frac{ 1}{ 64}$ $\frac{ 1}{ 16}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $1$
$0$ $1$ $1$ $1$
$0$ $2$ $2$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 1}{ 64}\)	\(\frac{ 1}{ 16}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

Hadamard product $I \ast \alpha$
A-Incarnation: diagonal subfamily of 1,1,1,1-intersection in $P^1 \times P^1 \times P^1 \times \P^1$
B-Incarnations:
Fibre products: 62211- x 632--1, S62211

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-128\)	\(96\)	\(48\)	\(16\)

2

New Number: 3.31 | AESZ: | Superseeker: 4 284 | Hash: 660b0951ad934fc17fda7eb9b1750649

Degree: 3

$\theta^4-2^{2} x(2\theta+1)^2(5\theta^2+5\theta+2)+2^{5} x^{2}(2\theta+1)(2\theta+3)(7\theta^2+14\theta+8)-2^{4} 11 x^{3}(2\theta+1)(2\theta+3)^2(2\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 168, 5360, 210280, ...
--> OEIS
Normalized instanton numbers (n₀=1): 4, 27, 284, 4368, 80968, ... ; Common denominator:...

Discriminant

$1-80z+896z^2-2816z^3$

No data for singularities

Note:

This is operator "3.31" from ...

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

3

New Number: 4.62 | AESZ: 292 | Superseeker: 4300/3 1701817028/3 | Hash: bce26f214ee56f65c7a275cd8fdcc0c7

Degree: 4

$3^{2} \theta^4-2^{2} 3 x\left(4636\theta^4+7928\theta^3+5347\theta^2+1383\theta+126\right)+2^{9} x^{2}\left(59048\theta^4+50888\theta^3-26248\theta^2-16827\theta-2205\right)-2^{16} 7 x^{3}\left(9004\theta^4-2304\theta^3-2511\theta^2-504\theta-27\right)-2^{24} 7^{2} x^{4}(4\theta+1)(2\theta+1)^2(4\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 168, 279720, 737721600, 2391487698600, ...
--> OEIS
Normalized instanton numbers (n₀=1): 4300/3, 1768292/3, 1701817028/3, 2484553593752/3, 1500880129466144, ... ; Common denominator:...

Discriminant

$-(65536z^2+5584z-1)(-3+896z)^2$

Local exponents

$-\frac{ 349}{ 8192}-\frac{ 85}{ 8192}\sqrt{ 17}$ $0$ $s_1$ $s_2$ $-\frac{ 349}{ 8192}+\frac{ 85}{ 8192}\sqrt{ 17}$ $\frac{ 3}{ 896}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 4}$
$1$ $0$ $1$ $1$ $1$ $1$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$ $1$ $3$ $\frac{ 1}{ 2}$
$2$ $0$ $2$ $2$ $2$ $4$ $\frac{ 3}{ 4}$

\(-\frac{ 349}{ 8192}-\frac{ 85}{ 8192}\sqrt{ 17}\)	\(0\)	\(s_1\)	\(s_2\)	\(-\frac{ 349}{ 8192}+\frac{ 85}{ 8192}\sqrt{ 17}\)	\(\frac{ 3}{ 896}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 4}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(3\)	\(\frac{ 1}{ 2}\)
\(2\)	\(0\)	\(2\)	\(2\)	\(2\)	\(4\)	\(\frac{ 3}{ 4}\)

Note:

Sporadic Operator.

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

4

New Number: 11.2 | AESZ: | Superseeker: 136/97 1768/97 | Hash: 940a6a9fb87fe9b9613bd73b990374c1

Degree: 11

\(97^{2} \theta^4+97 x\theta(1727\theta^3-2018\theta^2-1300\theta-291)-x^{2}\left(1652135\theta^4+13428812\theta^3+16174393\theta^2+10216234\theta+2709792\right)-3 x^{3}\left(27251145\theta^4+121375398\theta^3+189546499\theta^2+147705198\theta+46000116\right)-2 x^{4}\left(587751431\theta^4+2711697232\theta^3+5003189285\theta^2+4434707760\theta+1524637512\right)-x^{5}\left(9726250397\theta^4+50507429234\theta^3+106108023451\theta^2+103964102350\theta+38537290992\right)-2 3 x^{6}\left(8793822649\theta^4+52062405804\theta^3+122175610025\theta^2+130254629814\theta+51340027968\right)-2^{2} 3^{2} x^{7}\left(5429262053\theta^4+36477756530\theta^3+94431307279\theta^2+108363704338\theta+44982230808\right)-2^{4} 3^{2} x^{8}(\theta+1)(3432647479\theta^3+22487363787\theta^2+50808614711\theta+38959393614)-2^{4} 3^{3} x^{9}(\theta+1)(\theta+2)(1903493629\theta^2+10262864555\theta+14314039440)-2^{5} 3^{4} 13^{2} x^{10}(\theta+3)(\theta+2)(\theta+1)(1862987\theta+5992902)-2^{6} 3^{3} 13^{4} 7457 x^{11}(\theta+1)(\theta+2)(\theta+3)(\theta+4)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 18, 168, 2430, ...
--> OEIS
Normalized instanton numbers (n₀=1): 136/97, 292/97, 1768/97, 10128/97, 83387/97, ... ; Common denominator:...

Discriminant

$-(12z^2+6z+1)(7457z^5+6100z^4+1929z^3+257z^2+7z-1)(97+912z+2028z^2)^2$

Local exponents

$-\frac{ 38}{ 169}-\frac{ 1}{ 1014}\sqrt{ 2805}$ $-\frac{ 1}{ 4}-\frac{ 1}{ 12}\sqrt{ 3}I$ $-\frac{ 1}{ 4}+\frac{ 1}{ 12}\sqrt{ 3}I$ $-\frac{ 38}{ 169}+\frac{ 1}{ 1014}\sqrt{ 2805}$ $0$ $#ND+#NDI$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $1$ $0$ $1$ $2$
$3$ $1$ $1$ $3$ $0$ $1$ $3$
$4$ $2$ $2$ $4$ $0$ $2$ $4$

\(-\frac{ 38}{ 169}-\frac{ 1}{ 1014}\sqrt{ 2805}\)	\(-\frac{ 1}{ 4}-\frac{ 1}{ 12}\sqrt{ 3}I\)	\(-\frac{ 1}{ 4}+\frac{ 1}{ 12}\sqrt{ 3}I\)	\(-\frac{ 38}{ 169}+\frac{ 1}{ 1014}\sqrt{ 2805}\)	\(0\)	\(#ND+#NDI\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(2\)
\(3\)	\(1\)	\(1\)	\(3\)	\(0\)	\(1\)	\(3\)
\(4\)	\(2\)	\(2\)	\(4\)	\(0\)	\(2\)	\(4\)

Note:

This is operator "11.2" from ...

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

5

New Number: 8.40 | AESZ: | Superseeker: 5/4 35/2 | Hash: d3cb0fbbc65d6c5dace733d3d1ca181b

Degree: 8

$2^{4} \theta^4-2^{2} x\theta(2\theta^3+82\theta^2+53\theta+12)-x^{2}\left(4895\theta^4+18410\theta^3+26199\theta^2+18308\theta+5120\right)-x^{3}\left(60679\theta^4+272424\theta^3+497452\theta^2+430092\theta+143808\right)-x^{4}\left(344527\theta^4+1870838\theta^3+4034628\theta^2+3987101\theta+1478544\right)-x^{5}(\theta+1)(1076509\theta^3+5847783\theta^2+11226106\theta+7492832)-2 x^{6}(\theta+1)(\theta+2)(944887\theta^2+4249317\theta+5045304)-2^{8} 13 x^{7}(\theta+3)(\theta+2)(\theta+1)(518\theta+1381)-2^{5} 5 13^{2} 23 x^{8}(\theta+1)(\theta+2)(\theta+3)(\theta+4)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 20, 168, 2652, ...
--> OEIS
Normalized instanton numbers (n₀=1): 5/4, 57/16, 35/2, 459/4, 3615/4, ... ; Common denominator:...

Discriminant

$-(23z-1)(5z+1)(2z+1)(z+1)(13z+4)^2(4z+1)^2$

Local exponents

$-1$ $-\frac{ 1}{ 2}$ $-\frac{ 4}{ 13}$ $-\frac{ 1}{ 4}$ $-\frac{ 1}{ 5}$ $0$ $\frac{ 1}{ 23}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $\frac{ 1}{ 3}$ $1$ $0$ $1$ $2$
$1$ $1$ $3$ $\frac{ 2}{ 3}$ $1$ $0$ $1$ $3$
$2$ $2$ $4$ $1$ $2$ $0$ $2$ $4$

\(-1\)	\(-\frac{ 1}{ 2}\)	\(-\frac{ 4}{ 13}\)	\(-\frac{ 1}{ 4}\)	\(-\frac{ 1}{ 5}\)	\(0\)	\(\frac{ 1}{ 23}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 3}\)	\(1\)	\(0\)	\(1\)	\(2\)
\(1\)	\(1\)	\(3\)	\(\frac{ 2}{ 3}\)	\(1\)	\(0\)	\(1\)	\(3\)
\(2\)	\(2\)	\(4\)	\(1\)	\(2\)	\(0\)	\(2\)	\(4\)

Note:

This is operator "8.40" from ...

\(5+512\lambda\)	\(-1.-.620290874I\)	\(\frac{ 1}{ 3}+\frac{ 128}{ 3}\lambda\)	\(-\frac{ 1}{ 7}7^{ \frac{ 5}{ 8})exp(-\frac{ 9}{ 4}}Zeta(5)^(\frac{ 1}{ 4})-\frac{ 64}{ 3}\lambda\)
\(16\)	\(-3\)	\(\frac{ 4}{ 3}\)	\(-\frac{ 1}{ 3}-\frac{ 128}{ 3}\lambda\)
\(48\)	\(-12\)	\(5\)	\(-1.-.620290874I\)
\(192\)	\(-48\)	\(16\)	\(-3-512\lambda\)

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

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:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

\(1.+.620290874I\)	\(0\)	\(\frac{ 32}{ 3}\lambda\)	\(.8015849e-2\)
\(4\)	\(1\)	\(\frac{ 1}{ 3}\)	\(-\frac{ 32}{ 3}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(48\)	\(0\)	\(4\)	\(1.-.620290874I\)

\(-\frac{ 310145437}{ 500000000}I\)	\(12\)	\(1\)	\(1\)
\(-4\)	\(-24\)	\(-1\)	\(0\)
\(0\)	\(48\)	\(0\)	\(0\)
\(-48\)	\(0\)	\(0\)	\(0\)