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: 5.2 | AESZ: 19 | Superseeker: 80/23 4655/23 | Hash: 4532f44d62f644bf66aa7b153d4f5c5a

Degree: 5

$23^{2} \theta^4-23 x\left(921\theta^4+2046\theta^3+1644\theta^2+621\theta+92\right)-x^{2}\left(380851\theta^4+1328584\theta^3+1772673\theta^2+1033528\theta+221168\right)-2 x^{3}\left(475861\theta^4+1310172\theta^3+1028791\theta^2+208932\theta-27232\right)-2^{2} 17 x^{4}\left(8873\theta^4+14020\theta^3+5139\theta^2-1664\theta-976\right)+2^{3} 3 17^{2} x^{5}(\theta+1)^2(3\theta+2)(3\theta+4)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 4, 84, 2200, 71140, ...
--> OEIS
Normalized instanton numbers (n₀=1): 80/23, 1157/46, 4655/23, 71184/23, 1156690/23, ... ; Common denominator:...

Discriminant

$(54z-1)(z^2-11z-1)(23+34z)^2$

Local exponents

$-\frac{ 23}{ 34}$ $\frac{ 11}{ 2}-\frac{ 5}{ 2}\sqrt{ 5}$ $0$ $\frac{ 1}{ 54}$ $\frac{ 11}{ 2}+\frac{ 5}{ 2}\sqrt{ 5}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 2}{ 3}$
$1$ $1$ $0$ $1$ $1$ $1$
$3$ $1$ $0$ $1$ $1$ $1$
$4$ $2$ $0$ $2$ $2$ $\frac{ 4}{ 3}$

\(-\frac{ 23}{ 34}\)	\(\frac{ 11}{ 2}-\frac{ 5}{ 2}\sqrt{ 5}\)	\(0\)	\(\frac{ 1}{ 54}\)	\(\frac{ 11}{ 2}+\frac{ 5}{ 2}\sqrt{ 5}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 2}{ 3}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(3\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(4\)	\(2\)	\(0\)	\(2\)	\(2\)	\(\frac{ 4}{ 3}\)

Note:

This is operator "5.2" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-106\)	\(88\)	\(46\)	\(15\)

3

New Number: 5.3 | AESZ: 20 | Superseeker: 3 245/3 | Hash: a9a698dc5c79ffda497a7897390408b0

Degree: 5

$\theta^4-3 x\left(48\theta^4+60\theta^3+53\theta^2+23\theta+4\right)+3^{2} x^{2}\left(873\theta^4+1980\theta^3+2319\theta^2+1344\theta+304\right)-2 3^{4} x^{3}\left(1269\theta^4+3888\theta^3+5259\theta^2+3348\theta+800\right)+2^{2} 3^{6} x^{4}\left(891\theta^4+3240\theta^3+4653\theta^2+2952\theta+688\right)-2^{3} 3^{11} x^{5}(\theta+1)^2(3\theta+2)(3\theta+4)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 252, 6600, 198540, ...
--> OEIS
Normalized instanton numbers (n₀=1): 3, 33/2, 245/3, 879, 11829, ... ; Common denominator:...

Discriminant

$-(54z-1)(27z-1)^2(18z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 54}$ $\frac{ 1}{ 27}$ $\frac{ 1}{ 18}$ $\infty$
$0$ $0$ $0$ $0$ $\frac{ 2}{ 3}$
$0$ $1$ $\frac{ 1}{ 3}$ $1$ $1$
$0$ $1$ $\frac{ 2}{ 3}$ $3$ $1$
$0$ $2$ $1$ $4$ $\frac{ 4}{ 3}$

\(0\)	\(\frac{ 1}{ 54}\)	\(\frac{ 1}{ 27}\)	\(\frac{ 1}{ 18}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 2}{ 3}\)
\(0\)	\(1\)	\(\frac{ 1}{ 3}\)	\(1\)	\(1\)
\(0\)	\(1\)	\(\frac{ 2}{ 3}\)	\(3\)	\(1\)
\(0\)	\(2\)	\(1\)	\(4\)	\(\frac{ 4}{ 3}\)

Note:

A-Incarnation: (3,0),(0,3),(1,1) intersection in $P^3 \times \P^3$.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-18\)	\(72\)	\(54\)	\(15\)

4

New Number: 5.49 | AESZ: 248 | Superseeker: 7/3 148 | Hash: 0c9ccff1cb4f5096e455a9026799ed5a

Degree: 5

$3^{2} \theta^4-3 x\left(106\theta^4+146\theta^3+115\theta^2+42\theta+6\right)-x^{2}\left(4511\theta^4+24314\theta^3+37829\theta^2+23598\theta+5286\right)+2^{2} x^{3}\left(10457\theta^4+32184\theta^3+24449\theta^2+3627\theta-1317\right)-2^{2} 11 x^{4}\left(1596\theta^4+2040\theta^3-101\theta^2-1085\theta-386\right)-2^{4} 11^{2} x^{5}(4\theta+3)(\theta+1)^2(4\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 2, 54, 1028, 29110, ...
--> OEIS
Normalized instanton numbers (n₀=1): 7/3, 551/24, 148, 8241/4, 86854/3, ... ; Common denominator:...

Discriminant

$-(16z+1)(16z^2+44z-1)(-3+11z)^2$

Local exponents

$-\frac{ 11}{ 8}-\frac{ 5}{ 8}\sqrt{ 5}$ $-\frac{ 1}{ 16}$ $0$ $-\frac{ 11}{ 8}+\frac{ 5}{ 8}\sqrt{ 5}$ $\frac{ 3}{ 11}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 3}{ 4}$
$1$ $1$ $0$ $1$ $1$ $1$
$1$ $1$ $0$ $1$ $3$ $1$
$2$ $2$ $0$ $2$ $4$ $\frac{ 5}{ 4}$

\(-\frac{ 11}{ 8}-\frac{ 5}{ 8}\sqrt{ 5}\)	\(-\frac{ 1}{ 16}\)	\(0\)	\(-\frac{ 11}{ 8}+\frac{ 5}{ 8}\sqrt{ 5}\)	\(\frac{ 3}{ 11}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 3}{ 4}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(0\)	\(1\)	\(3\)	\(1\)
\(2\)	\(2\)	\(0\)	\(2\)	\(4\)	\(\frac{ 5}{ 4}\)

Note:

This is operator "5.49" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-92\)	\(84\)	\(48\)	\(15\)

5

New Number: 8.16 | AESZ: 196 | Superseeker: 189/47 9277/47 | Hash: fdeee36c14d9c003b59c1738c024d479

Degree: 8

$47^{2} \theta^4-47 x\left(2489\theta^4+4984\theta^3+4043\theta^2+1551\theta+235\right)-x^{2}\left(161022+701851\theta+1135848\theta^2+790072\theta^3+208867\theta^4\right)+x^{3}\left(38352+149319\theta+383912\theta^2+637644\theta^3+370857\theta^4\right)-x^{4}\left(1770676+5161283\theta+4424049\theta^2+511820\theta^3-291161\theta^4\right)+x^{5}\left(2151-260936\theta-750755\theta^2-749482\theta^3-406192\theta^4\right)+3^{3} x^{6}\left(5305\theta^4+90750\theta^3+152551\theta^2+91194\theta+17914\right)+2 3^{6} x^{7}\left(106\theta^4+230\theta^3+197\theta^2+82\theta+15\right)-2^{2} 3^{10} x^{8}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 5, 93, 2507, 81229, ...
--> OEIS
Normalized instanton numbers (n₀=1): 189/47, 979/47, 9277/47, 124795/47, 2049020/47, ... ; Common denominator:...

Discriminant

$-(-1+53z+90z^2-50z^3+81z^4)(-47-z+54z^2)^2$

Local exponents

$\frac{ 1}{ 108}-\frac{ 1}{ 108}\sqrt{ 10153}$ $0$ $\frac{ 1}{ 108}+\frac{ 1}{ 108}\sqrt{ 10153}$ $#ND+#NDI$ $\infty$
$0$ $0$ $0$ $0$ $1$
$1$ $0$ $1$ $1$ $1$
$3$ $0$ $3$ $1$ $1$
$4$ $0$ $4$ $2$ $1$

\(\frac{ 1}{ 108}-\frac{ 1}{ 108}\sqrt{ 10153}\)	\(0\)	\(\frac{ 1}{ 108}+\frac{ 1}{ 108}\sqrt{ 10153}\)	\(#ND+#NDI\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(3\)	\(0\)	\(3\)	\(1\)	\(1\)
\(4\)	\(0\)	\(4\)	\(2\)	\(1\)

Note:

The operator has a second MUM-point at infinity, corresponding to operator 8.17 .

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-90\)	\(86\)	\(47\)	\(15\)

6

New Number: 8.24 | AESZ: 286 | Superseeker: 3 437/3 | Hash: 94afcd38a40c3a3e54fc3c57b4b85459

Degree: 8

$3^{2} \theta^4-3^{2} x\left(38\theta^4+82\theta^3+67\theta^2+26\theta+4\right)-3 x^{2}\left(2045\theta^4+5702\theta^3+7535\theta^2+4170\theta+852\right)+2^{3} 3 x^{3}\left(2208\theta^4+5925\theta^3+7925\theta^2+5607\theta+1512\right)+2^{3} x^{4}\left(60287\theta^4+56374\theta^3-215983\theta^2-268986\theta-85452\right)-2^{4} x^{5}\left(205651\theta^4+605608\theta^3+603579\theta^2+204622\theta+8104\right)-2^{7} x^{6}\left(51414\theta^4-273267\theta^3-502700\theta^2-305649\theta-63398\right)+2^{8} 37 x^{7}\left(7909\theta^4+18122\theta^3+17595\theta^2+8462\theta+1672\right)-2^{13} 37^{2} x^{8}(4\theta+3)(\theta+1)^2(4\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 4, 72, 1696, 49960, ...
--> OEIS
Normalized instanton numbers (n₀=1): 3, 539/24, 437/3, 18531/8, 90274/3, ... ; Common denominator:...

Discriminant

$-(-1+40z+504z^2-3088z^3+8192z^4)(-3-3z+148z^2)^2$

Local exponents

$\frac{ 3}{ 296}-\frac{ 1}{ 296}\sqrt{ 1785}$ ≈$-0.070843$ $0$ ≈$0.020383$ $\frac{ 3}{ 296}+\frac{ 1}{ 296}\sqrt{ 1785}$ ≈$0.213707$ ≈$0.213707$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 3}{ 4}$
$1$ $1$ $0$ $1$ $1$ $1$ $1$ $1$
$3$ $1$ $0$ $1$ $3$ $1$ $1$ $1$
$4$ $2$ $0$ $2$ $4$ $2$ $2$ $\frac{ 5}{ 4}$

\(\frac{ 3}{ 296}-\frac{ 1}{ 296}\sqrt{ 1785}\)	≈\(-0.070843\)	\(0\)	≈\(0.020383\)	\(\frac{ 3}{ 296}+\frac{ 1}{ 296}\sqrt{ 1785}\)	≈\(0.213707\)	≈\(0.213707\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 3}{ 4}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(3\)	\(1\)	\(0\)	\(1\)	\(3\)	\(1\)	\(1\)	\(1\)
\(4\)	\(2\)	\(0\)	\(2\)	\(4\)	\(2\)	\(2\)	\(\frac{ 5}{ 4}\)

Note:

This is operator "8.24" from ...

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

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

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

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

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