Calabi-Yau differential operator database v.3.0

1

New Number: 5.10 | AESZ: 59 | Superseeker: 30/7 124 | Hash: f47563daeb0f7328bd675f13cfb84a55

Degree: 5

$7^{2} \theta^4-2 7 x\left(257\theta^4+520\theta^3+435\theta^2+175\theta+28\right)+2^{2} x^{2}\left(13497\theta^4+55536\theta^3+81222\theta^2+50337\theta+11396\right)-2^{3} x^{3}\left(17201\theta^4+114996\theta^3+248466\theta^2+202629\theta+55412\right)-2^{4} x^{4}\left(5762\theta^4+29668\theta^3+48150\theta^2+31741\theta+7412\right)-2^{5} 3 x^{5}(4\theta+5)(3\theta+2)(3\theta+4)(4\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 144, 3680, 114400, ...
--> OEIS
Normalized instanton numbers (n₀=1): 30/7, 129/14, 124, 72129/56, 130434/7, ... ; Common denominator:...

Discriminant

$-(4z-1)(16z-1)(54z-1)(7+2z)^2$

Local exponents

$-\frac{ 7}{ 2}$ $0$ $\frac{ 1}{ 54}$ $\frac{ 1}{ 16}$ $\frac{ 1}{ 4}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 2}{ 3}$
$1$ $0$ $1$ $1$ $1$ $\frac{ 3}{ 4}$
$3$ $0$ $1$ $1$ $1$ $\frac{ 5}{ 4}$
$4$ $0$ $2$ $2$ $2$ $\frac{ 4}{ 3}$

\(-\frac{ 7}{ 2}\)	\(0\)	\(\frac{ 1}{ 54}\)	\(\frac{ 1}{ 16}\)	\(\frac{ 1}{ 4}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 2}{ 3}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 3}{ 4}\)
\(3\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 5}{ 4}\)
\(4\)	\(0\)	\(2\)	\(2\)	\(2\)	\(\frac{ 4}{ 3}\)

Note:

This is operator "5.10" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-78\)	\(92\)	\(56\)	\(17\)

2

New Number: 5.20 | AESZ: 186 | Superseeker: 49/19 1761/19 | Hash: b3d164f22d02de1efcd62d3aa9ab5ce4

Degree: 5

$19^{2} \theta^4-19 x\left(700\theta^4+1238\theta^3+999\theta^2+380\theta+57\right)-x^{2}\left(64745\theta^4+368006\theta^3+609133\theta^2+412756\theta+102258\right)+3^{3} x^{3}\left(6397\theta^4+12198\theta^3-11923\theta^2-27360\theta-11286\right)+3^{6} x^{4}\left(64\theta^4+1154\theta^3+2425\theta^2+1848\theta+486\right)-3^{11} x^{5}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 3, 51, 1029, 25299, ...
--> OEIS
Normalized instanton numbers (n₀=1): 49/19, 252/19, 1761/19, 18990/19, 246159/19, ... ; Common denominator:...

Discriminant

$-(z+1)(243z^2+35z-1)(-19+27z)^2$

Local exponents

$-1$ $-\frac{ 35}{ 486}-\frac{ 13}{ 486}\sqrt{ 13}$ $0$ $-\frac{ 35}{ 486}+\frac{ 13}{ 486}\sqrt{ 13}$ $\frac{ 19}{ 27}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $0$ $1$ $1$ $1$
$1$ $1$ $0$ $1$ $3$ $1$
$2$ $2$ $0$ $2$ $4$ $1$

\(-1\)	\(-\frac{ 35}{ 486}-\frac{ 13}{ 486}\sqrt{ 13}\)	\(0\)	\(-\frac{ 35}{ 486}+\frac{ 13}{ 486}\sqrt{ 13}\)	\(\frac{ 19}{ 27}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(0\)	\(1\)	\(3\)	\(1\)
\(2\)	\(2\)	\(0\)	\(2\)	\(4\)	\(1\)

Note:

There is a second MUM-point at infinity, corresponding to Operator AESZ 187/5.21

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-84\)	\(90\)	\(57\)	\(17\)

3

New Number: 5.80 | AESZ: 311 | Superseeker: 25/13 875/13 | Hash: 8219f3f4bd56f6c2b2cc3ab9093b65d1

Degree: 5

$13^{2} \theta^4-13 x\left(327\theta^4+1038\theta^3+857\theta^2+338\theta+52\right)-2^{4} x^{2}\left(12848\theta^4+42008\theta^3+52082\theta^2+28548\theta+5707\right)-2^{11} x^{3}\left(122\theta^4-1872\theta^3-6341\theta^2-5772\theta-1547\right)+2^{16} x^{4}(2\theta+1)(76\theta^3+426\theta^2+570\theta+227)+2^{23} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 4, 84, 1840, 56980, ...
--> OEIS
Normalized instanton numbers (n₀=1): 25/13, 1359/52, 875/13, 36572/13, 256800/13, ... ; Common denominator:...

Discriminant

$(8192z^3-896z^2-35z+1)(13+64z)^2$

Local exponents

$-\frac{ 13}{ 64}$ ≈$-0.045147$ $0$ ≈$0.020117$ ≈$0.134405$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $1$ $0$ $1$ $1$ $1$
$3$ $1$ $0$ $1$ $1$ $1$
$4$ $2$ $0$ $2$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 13}{ 64}\)	≈\(-0.045147\)	\(0\)	≈\(0.020117\)	≈\(0.134405\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(3\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(4\)	\(2\)	\(0\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.80" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-144\)	\(100\)	\(52\)	\(17\)

4

New Number: 8.9 | AESZ: 174 | Superseeker: 16 -13 | Hash: 3f987b46d9ebf201eeead1a885b78e66

Degree: 8

$\theta^4-x(11\theta^2+11\theta+3)(17\theta^2+17\theta+6)+x^{2}\left(8711\theta^4+33980\theta^3+47095\theta^2+26230\theta+5232\right)-2^{3} 3^{2} x^{3}\left(187\theta^4-1122\theta^3-3436\theta^2-2595\theta-684\right)+2^{4} 3^{2} x^{4}\left(8639\theta^4+17278\theta^3-11650\theta^2-20289\theta-6102\right)+2^{6} 3^{4} x^{5}\left(187\theta^4+1870\theta^3+1052\theta^2-163\theta-216\right)+2^{6} 3^{4} x^{6}\left(8711\theta^4+864\theta^3-2579\theta^2+864\theta+828\right)+2^{9} 3^{6} x^{7}(11\theta^2+11\theta+3)(17\theta^2+17\theta+6)+2^{12} 3^{8} x^{8}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 18, 798, 45864, 2994894, ...
--> OEIS
Normalized instanton numbers (n₀=1): 16, 7/2, -13, 11663/2, -26414, ... ; Common denominator:...

Discriminant

$(81z^2+99z-1)(64z^2+88z-1)(1+72z^2)^2$

Local exponents

$-\frac{ 11}{ 16}-\frac{ 5}{ 16}\sqrt{ 5}$ $-\frac{ 11}{ 18}-\frac{ 5}{ 18}\sqrt{ 5}$ $0-\frac{ 1}{ 12}\sqrt{ 2}I$ $0$ $0+\frac{ 1}{ 12}\sqrt{ 2}I$ $-\frac{ 11}{ 18}+\frac{ 5}{ 18}\sqrt{ 5}$ $-\frac{ 11}{ 16}+\frac{ 5}{ 16}\sqrt{ 5}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $0$ $1$ $1$ $1$ $1$
$1$ $1$ $3$ $0$ $3$ $1$ $1$ $1$
$2$ $2$ $4$ $0$ $4$ $2$ $2$ $1$

\(-\frac{ 11}{ 16}-\frac{ 5}{ 16}\sqrt{ 5}\)	\(-\frac{ 11}{ 18}-\frac{ 5}{ 18}\sqrt{ 5}\)	\(0-\frac{ 1}{ 12}\sqrt{ 2}I\)	\(0\)	\(0+\frac{ 1}{ 12}\sqrt{ 2}I\)	\(-\frac{ 11}{ 18}+\frac{ 5}{ 18}\sqrt{ 5}\)	\(-\frac{ 11}{ 16}+\frac{ 5}{ 16}\sqrt{ 5}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(3\)	\(0\)	\(3\)	\(1\)	\(1\)	\(1\)
\(2\)	\(2\)	\(4\)	\(0\)	\(4\)	\(2\)	\(2\)	\(1\)

Note:

Hadamard product $ b \ast g$. This operator has a second MUM-point at infinity with the same instanton numbers. It
can be reduced to an operator of degree 4 with a single MUM-point defined over $Q(\sqrt{?})$.

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:

Characteristic classes:

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:

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice