Calabi-Yau differential operator database v.3.0

1

New Number: 2.25 | AESZ: 138 | Superseeker: 27 2618 | Hash: c524254b716132352b27914640b03c8b

Degree: 2

$\theta^4-3 x(3\theta+1)(3\theta+2)(17\theta^2+17\theta+6)+2^{3} 3^{4} x^{2}(3\theta+1)(3\theta+2)(3\theta+4)(3\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 36, 3780, 524160, 82952100, ...
--> OEIS
Normalized instanton numbers (n₀=1): 27, 189/4, 2618, 43713, 2319057, ... ; Common denominator:...

Discriminant

$(243z-1)(216z-1)$

Local exponents

$0$ $\frac{ 1}{ 243}$ $\frac{ 1}{ 216}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 3}$
$0$ $1$ $1$ $\frac{ 2}{ 3}$
$0$ $1$ $1$ $\frac{ 4}{ 3}$
$0$ $2$ $2$ $\frac{ 5}{ 3}$

\(0\)	\(\frac{ 1}{ 243}\)	\(\frac{ 1}{ 216}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 2}{ 3}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 4}{ 3}\)
\(0\)	\(2\)	\(2\)	\(\frac{ 5}{ 3}\)

Note:

Hadamard product $B\ast g$.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-156\)	\(108\)	\(36\)	\(15\)

2

New Number: 2.56 | AESZ: 185 | Superseeker: 6 608 | Hash: 80506439e4d4fdc41f5b16e246a69fbf

Degree: 2

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 6, 162, 6180, 284130, ...
--> OEIS
Normalized instanton numbers (n₀=1): 6, 93/2, 608, 11754, 275352, ... ; Common denominator:...

Discriminant

$1-72z-432z^2$

Local exponents

$-\frac{ 1}{ 12}-\frac{ 1}{ 18}\sqrt{ 3}$ $0$ $-\frac{ 1}{ 12}+\frac{ 1}{ 18}\sqrt{ 3}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$
$1$ $0$ $1$ $1$
$2$ $0$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 1}{ 12}-\frac{ 1}{ 18}\sqrt{ 3}\)	\(0\)	\(-\frac{ 1}{ 12}+\frac{ 1}{ 18}\sqrt{ 3}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)
\(1\)	\(0\)	\(1\)	\(1\)
\(2\)	\(0\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

Hadamard product $I \ast \zeta$

	Gopakumar-Vafa invariants
g=0	216, 1674, 21888, 423144, 9912672, 265889088, 7814158560, 245812722408,...
g=1	0, 0, 0, 0, 432, 374244, 47016720, 3817401912,...
g=2	,...

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-120\)	\(84\)	\(36\)	\(13\)

3

New Number: 3.10 | AESZ: ~103 | Superseeker: 10 664 | Hash: 9239615e8ac132ca232c13367a39ae3b

Degree: 3

$\theta^4-2 x\left(86\theta^4+172\theta^3+143\theta^2+57\theta+9\right)+2^{2} 3^{2} x^{2}(\theta+1)^2(236\theta^2+472\theta+187)-2^{4} 3^{4} 5^{2} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 18, 630, 28980, 1593270, ...
--> OEIS
Normalized instanton numbers (n₀=1): 10, 24, 664, 9088, 234388, ... ; Common denominator:...

Discriminant

$-(100z-1)(-1+36z)^2$

Local exponents

$0$ $\frac{ 1}{ 100}$ $\frac{ 1}{ 36}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $\frac{ 1}{ 2}$ $1$
$0$ $1$ $\frac{ 1}{ 2}$ $2$
$0$ $2$ $1$ $\frac{ 5}{ 2}$

\(0\)	\(\frac{ 1}{ 100}\)	\(\frac{ 1}{ 36}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(\frac{ 1}{ 2}\)	\(1\)
\(0\)	\(1\)	\(\frac{ 1}{ 2}\)	\(2\)
\(0\)	\(2\)	\(1\)	\(\frac{ 5}{ 2}\)

Note:

Operator equivalent to AESZ 103 =$c \ast c$.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-48\)	\(72\)	\(36\)	\(12\)

4

New Number: 3.8 | AESZ: ~100 | Superseeker: 5 454 | Hash: 82a1ac6ac6fb9ab2e4d6b5d5790d1d9b

Degree: 3

$\theta^4+x\left(15\theta^4+30\theta^3+35\theta^2+20\theta+4\right)-2^{5} x^{2}(\theta+1)^2(66\theta^2+132\theta+53)-2^{8} 7^{2} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, -4, 132, -1120, 72100, ...
--> OEIS
Normalized instanton numbers (n₀=1): 5, 42, 454, 7498, 154351, ... ; Common denominator:...

Discriminant

$-(49z-1)(1+32z)^2$

Local exponents

$-\frac{ 1}{ 32}$ $0$ $\frac{ 1}{ 49}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$\frac{ 1}{ 2}$ $0$ $1$ $1$
$\frac{ 1}{ 2}$ $0$ $1$ $2$
$1$ $0$ $2$ $\frac{ 5}{ 2}$

\(-\frac{ 1}{ 32}\)	\(0\)	\(\frac{ 1}{ 49}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(\frac{ 1}{ 2}\)	\(0\)	\(1\)	\(1\)
\(\frac{ 1}{ 2}\)	\(0\)	\(1\)	\(2\)
\(1\)	\(0\)	\(2\)	\(\frac{ 5}{ 2}\)

Note:

Operator equivalent to AESZ 100= $ a \ast a$

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-72\)	\(72\)	\(36\)	\(12\)

5

New Number: 5.100 | AESZ: 347 | Superseeker: 15 27140/3 | Hash: f00de20026c099e75b447c475ab287e4

Degree: 5

$\theta^4-3 x\left(213\theta^4+186\theta^3+149\theta^2+56\theta+8\right)+2^{3} 3^{3} x^{2}\left(702\theta^4+1078\theta^3+949\theta^2+392\theta+60\right)-2^{6} 3^{3} x^{3}\left(9277\theta^4+18432\theta^3+16008\theta^2+6000\theta+840\right)+2^{13} 3^{4} 5 x^{4}(2\theta+1)^2(51\theta^2+69\theta+32)-2^{14} 3^{6} 5^{2} x^{5}(2\theta+1)^2(2\theta+3)^2$

Maple LaTex

Coefficients of the holomorphic solution: 1, 24, 1944, 218400, 28488600, ...
--> OEIS
Normalized instanton numbers (n₀=1): 15, 1329/4, 27140/3, 220680, 5952570, ... ; Common denominator:...

Discriminant

$-(192z-1)(1728z^2-207z+1)(-1+120z)^2$

Local exponents

$0$ $\frac{ 23}{ 384}-\frac{ 11}{ 1152}\sqrt{ 33}$ $\frac{ 1}{ 192}$ $\frac{ 1}{ 120}$ $\frac{ 23}{ 384}+\frac{ 11}{ 1152}\sqrt{ 33}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $1$ $1$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $3$ $1$ $\frac{ 3}{ 2}$
$0$ $2$ $2$ $4$ $2$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 23}{ 384}-\frac{ 11}{ 1152}\sqrt{ 33}\)	\(\frac{ 1}{ 192}\)	\(\frac{ 1}{ 120}\)	\(\frac{ 23}{ 384}+\frac{ 11}{ 1152}\sqrt{ 33}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(3\)	\(1\)	\(\frac{ 3}{ 2}\)
\(0\)	\(2\)	\(2\)	\(4\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.100" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-16\)	\(84\)	\(36\)	\(13\)

6

New Number: 5.87 | AESZ: 321 | Superseeker: 35/9 3002/9 | Hash: b786027c217dd5d5c5abac7b1ecc570b

Degree: 5

$3^{4} \theta^4-3^{2} x\left(191\theta^4+862\theta^3+683\theta^2+252\theta+36\right)-2^{5} x^{2}\left(7225\theta^4+24835\theta^3+30634\theta^2+16173\theta+3069\right)-2^{8} x^{3}\left(13251\theta^4+35856\theta^3+27641\theta^2+6966\theta+180\right)-2^{12} 5 x^{4}(2\theta+1)(314\theta^3+363\theta^2+68\theta-31)+2^{16} 5^{2} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 4, 132, 4000, 179620, ...
--> OEIS
Normalized instanton numbers (n₀=1): 35/9, 261/4, 3002/9, 126800/9, 1727129/9, ... ; Common denominator:...

Discriminant

$(32z+1)(32z^2-71z+1)(9+80z)^2$

Local exponents

$-\frac{ 9}{ 80}$ $-\frac{ 1}{ 32}$ $0$ $\frac{ 71}{ 64}-\frac{ 17}{ 64}\sqrt{ 17}$ $\frac{ 71}{ 64}+\frac{ 17}{ 64}\sqrt{ 17}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $1$ $0$ $1$ $1$ $1$
$3$ $1$ $0$ $1$ $1$ $1$
$4$ $2$ $0$ $2$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 9}{ 80}\)	\(-\frac{ 1}{ 32}\)	\(0\)	\(\frac{ 71}{ 64}-\frac{ 17}{ 64}\sqrt{ 17}\)	\(\frac{ 71}{ 64}+\frac{ 17}{ 64}\sqrt{ 17}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(3\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(4\)	\(2\)	\(0\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.87" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-128\)	\(84\)	\(36\)	\(13\)

7

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 $Q(\sqrt{-2})$

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-60\)	\(72\)	\(36\)	\(12\)

8

New Number: 8.30 | AESZ: 314 | Superseeker: 229/4 297111/4 | Hash: 893692ba7eb3effcbc0c3b48d405456a

Degree: 8

$2^{4} \theta^4-2^{2} x\left(1282\theta^4+2618\theta^3+1909\theta^2+600\theta+72\right)-3^{2} x^{2}\left(9503\theta^4+26810\theta^3+31755\theta^2+15944\theta+2936\right)+3^{4} x^{3}\left(15627\theta^4-18288\theta^3-91412\theta^2-53256\theta-9688\right)+2 3^{6} x^{4}\left(15106\theta^4+20300\theta^3-20421\theta^2-23443\theta-5907\right)-2^{2} 3^{8} x^{5}\left(2072\theta^4-18256\theta^3-2563\theta^2+4626\theta+1495\right)-2^{2} 3^{10} x^{6}\left(6204\theta^4+360\theta^3-281\theta^2+1017\theta+434\right)-2^{5} 3^{12} x^{7}(2\theta+1)(100\theta^3+162\theta^2+95\theta+21)+2^{8} 3^{14} x^{8}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 18, 1926, 310860, 61060230, ...
--> OEIS
Normalized instanton numbers (n₀=1): 229/4, 1293, 297111/4, 6150238, 2540085295/4, ... ; Common denominator:...

Discriminant

$(z-1)(11664z^3+3888z^2+324z-1)(-4-9z+648z^2)^2$

Local exponents

≈$-0.168156-0.022431I$ ≈$-0.168156+0.022431I$ $\frac{ 1}{ 144}-\frac{ 1}{ 144}\sqrt{ 129}$ $0$ $\frac{ 1}{ 18}2^(\frac{ 1}{ 3})+\frac{ 1}{ 36}2^(\frac{ 2}{ 3})-\frac{ 1}{ 9}$ $\frac{ 1}{ 144}+\frac{ 1}{ 144}\sqrt{ 129}$ $1$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $1$ $1$ $0$ $1$ $1$ $1$ $1$
$1$ $1$ $3$ $0$ $1$ $3$ $1$ $1$
$2$ $2$ $4$ $0$ $2$ $4$ $2$ $\frac{ 3}{ 2}$

≈\(-0.168156-0.022431I\)	≈\(-0.168156+0.022431I\)	\(\frac{ 1}{ 144}-\frac{ 1}{ 144}\sqrt{ 129}\)	\(0\)	\(\frac{ 1}{ 18}2^(\frac{ 1}{ 3})+\frac{ 1}{ 36}2^(\frac{ 2}{ 3})-\frac{ 1}{ 9}\)	\(\frac{ 1}{ 144}+\frac{ 1}{ 144}\sqrt{ 129}\)	\(1\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(3\)	\(0\)	\(1\)	\(3\)	\(1\)	\(1\)
\(2\)	\(2\)	\(4\)	\(0\)	\(2\)	\(4\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "8.30" from ...

\(.84386396e-1+.881341465I\)	\(-.593907697+1.816219110I\)	\(.987405021+.251865740I\)	\(-.18197034e-1-.280246794I\)
\(-.610146073+4.105015738I\)	\(3.086102688+5.881005017I\)	\(\frac{ 1}{ ln}(2)^{ \frac{ 8}{ 3}}ln(3)^{ \frac{ 8}{ 5}}Zeta(5)^8-1.317777791I\)	\(-.605925748-.688412508I\)
\(-1.084750363+4.722447485I\)	\(1.891556916+7.035664131I\)	\(4.676916900-1.255046419I\)	\(-.642466556-.856579334I\)
\(-4.881168587+32.840125900I\)	\(16.688821505+47.048040140I\)	\(24.445126799-10.542222328I\)	\(-3.847405984-5.507300063I\)

\(8+480\lambda\)	\(-\frac{ 7}{ 2}-240\lambda\)	\(\frac{ 91}{ 72}+\frac{ 260}{ 3}\lambda\)	\(-.302703484-\frac{ 140}{ 3}\lambda\)
\(26\)	\(-12\)	\(\frac{ 169}{ 36}\)	\(-\frac{ 91}{ 72}-\frac{ 260}{ 3}\lambda\)
\(72\)	\(-36\)	\(14\)	\(-\frac{ 7}{ 2}-240\lambda\)
\(144\)	\(-72\)	\(26\)	\(-6-480\lambda\)

\(1.+.755979502I\)	\(0\)	\(\frac{ 39}{ 2}\lambda\)	\(.15875139e-1\)
\(\frac{ 9}{ 2}\)	\(1\)	\(\frac{ 9}{ 16}\)	\(-\frac{ 39}{ 2}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(36\)	\(0\)	\(\frac{ 9}{ 2}\)	\(1.-.755979502I\)

\(-\frac{ 377989751}{ 500000000}I\)	\(\frac{ 21}{ 2}\)	\(1\)	\(1\)
\(-\frac{ 9}{ 2}\)	\(-18\)	\(-1\)	\(0\)
\(0\)	\(36\)	\(0\)	\(0\)
\(-36\)	\(0\)	\(0\)	\(0\)

\(1+120\lambda\)	\(0\)	\(\frac{ 35}{ 3}\lambda\)	\(.9393573e-2\)
\(\frac{ 7}{ 2}\)	\(1\)	\(\frac{ 49}{ 144}\)	\(-\frac{ 35}{ 3}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(36\)	\(0\)	\(\frac{ 7}{ 2}\)	\(1-120\lambda\)

\(-120\lambda\)	\(\frac{ 19}{ 2}\)	\(1\)	\(1\)
\(-\frac{ 7}{ 2}\)	\(-18\)	\(-1\)	\(0\)
\(0\)	\(36\)	\(0\)	\(0\)
\(-36\)	\(0\)	\(0\)	\(0\)

\(-48\lambda\)	\(9\)	\(1\)	\(1\)
\(-3\)	\(-18\)	\(-1\)	\(0\)
\(0\)	\(36\)	\(0\)	\(0\)
\(-36\)	\(0\)	\(0\)	\(0\)

Calabi-Yau differential operator database v.3

Summary

Discriminant

Local exponents

Note:

Explicit solution

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

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