Calabi-Yau differential operator database v.3.0

1

New Number: 5.1 | AESZ: 17 | Superseeker: 6/5 118/5 | Hash: 370d10edbf5900002f79cf6163e106a5

Degree: 5

$5^{2} \theta^4-3 5 x\left(51\theta^4+84\theta^3+72\theta^2+30\theta+5\right)+2 3 x^{2}\left(531\theta^4+828\theta^3+541\theta^2+155\theta+15\right)-2 3^{3} x^{3}\left(423\theta^4+2160\theta^3+4399\theta^2+3795\theta+1170\right)+3^{5} x^{4}\left(279\theta^4+1368\theta^3+2270\theta^2+1586\theta+402\right)-3^{10} x^{5}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 3, 27, 381, 6219, ...
--> OEIS
Normalized instanton numbers (n₀=1): 6/5, 39/10, 118/5, 1443/10, 6108/5, ... ; Common denominator:...

Discriminant

$-(27z-1)(27z^2+1)(-5+9z)^2$

Local exponents

$0-\frac{ 1}{ 9}\sqrt{ 3}I$ $0$ $0+\frac{ 1}{ 9}\sqrt{ 3}I$ $\frac{ 1}{ 27}$ $\frac{ 5}{ 9}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $1$
$1$ $0$ $1$ $1$ $1$ $1$
$1$ $0$ $1$ $1$ $3$ $1$
$2$ $0$ $2$ $2$ $4$ $1$

\(0-\frac{ 1}{ 9}\sqrt{ 3}I\)	\(0\)	\(0+\frac{ 1}{ 9}\sqrt{ 3}I\)	\(\frac{ 1}{ 27}\)	\(\frac{ 5}{ 9}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)
\(1\)	\(0\)	\(1\)	\(1\)	\(3\)	\(1\)
\(2\)	\(0\)	\(2\)	\(2\)	\(4\)	\(1\)

Note:

There is a second MUM-point at infinity,
corresponding to Operator AESZ 290/5.71
A-Incarnation: diagonal subfamily 1,1,1-section in $P^2 \times P^2 \times P^2$

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-90\)	\(108\)	\(90\)	\(24\)

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: 10.1 | AESZ: | Superseeker: 118/91 268/13 | Hash: 9708eba070b10afbba48d1f539423c22

Degree: 10

$7^{2} 13^{2} \theta^4-7 13 x\left(2221\theta^4+4604\theta^3+3940\theta^2+1638\theta+273\right)-2 x^{2}\left(275775\theta^4+850032\theta^3+1167211\theta^2+754481\theta+190918\right)+x^{3}\left(27353\theta^4-6829166\theta^2-6586125\theta-2489994\theta^3-2242968\right)-x^{4}\left(46728731\theta+12063734\theta^3+18386820+508804\theta^4+40173426\theta^2\right)+3 x^{5}\left(33450\theta^4+319414\theta^3-766536\theta^2-1551527\theta-668977\right)+x^{6}\left(2892684+47526449\theta^2+4076796\theta^4+26519901\theta+28614978\theta^3\right)-2 x^{7}\left(96271\theta^4+1136261\theta^3+4541506\theta^2+6411261\theta+2925345\right)-13 x^{8}(\theta+1)(257369\theta^3+699321\theta^2+523184\theta+25156)+2^{2} 5 13^{2} x^{9}(\theta+2)(\theta+1)(227\theta^2+762\theta+681)-2^{2} 5^{2} 13^{3} x^{10}(\theta+1)(\theta+2)^2(\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 3, 29, 393, 6333, ...
--> OEIS
Normalized instanton numbers (n₀=1): 118/91, 373/91, 268/13, 12732/91, 105020/91, ... ; Common denominator:...

Discriminant

$-(-1+25z+49z^2+36z^3+199z^4-40z^5+13z^6)(-91-27z+130z^2)^2$

Local exponents

$\frac{ 27}{ 260}-\frac{ 1}{ 260}\sqrt{ 48049}$ $0$ $\frac{ 27}{ 260}+\frac{ 1}{ 260}\sqrt{ 48049}$ $#ND+#NDI$ $\infty$
$0$ $0$ $0$ $0$ $1$
$1$ $0$ $1$ $1$ $2$
$3$ $0$ $3$ $1$ $2$
$4$ $0$ $4$ $2$ $3$

\(\frac{ 27}{ 260}-\frac{ 1}{ 260}\sqrt{ 48049}\)	\(0\)	\(\frac{ 27}{ 260}+\frac{ 1}{ 260}\sqrt{ 48049}\)	\(#ND+#NDI\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(0\)	\(1\)	\(1\)	\(2\)
\(3\)	\(0\)	\(3\)	\(1\)	\(2\)
\(4\)	\(0\)	\(4\)	\(2\)	\(3\)

Note:

This is operator "10.1" from ...

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

4

New Number: 11.19 | AESZ: | Superseeker: 21/4 -1045/6 | Hash: acf903f94ac2a08b9f2b26dff65a52ff

Degree: 11

$2^{4} \theta^4-2^{2} 3 x\left(42\theta^4+102\theta^3+79\theta^2+28\theta+4\right)+3^{3} x^{2}\left(315\theta^4+4266\theta^3+5903\theta^2+3052\theta+596\right)+3^{6} x^{3}\left(1318\theta^4+2322\theta^3+1973\theta^2+1480\theta+380\right)-3^{8} x^{4}\left(929\theta^4-9440\theta^3-49249\theta^2-40585\theta-10625\right)-3^{10} x^{5}\left(2379\theta^4-3180\theta^3+21452\theta^2+12663\theta+2214\right)-3^{12} x^{6}\left(1180\theta^4+108042\theta^3+173091\theta^2+112103\theta+25380\right)+3^{13} x^{7}\left(27312\theta^4+432678\theta^3+80098\theta^2-241649\theta-108332\right)+3^{15} x^{8}\left(107658\theta^4+438498\theta^3+811975\theta^2+529736\theta+119035\right)-3^{18} x^{9}\left(96469\theta^4+336390\theta^3+294983\theta^2+82398\theta+582\right)+2 3^{20} 13 x^{10}\left(2902\theta^4+5462\theta^3+3737\theta^2+1006\theta+63\right)+2^{2} 3^{23} 13^{2} x^{11}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 3, -27, -1563, -40491, ...
--> OEIS
Normalized instanton numbers (n₀=1): 21/4, -969/16, -1045/6, -35199/4, 536619/4, ... ; Common denominator:...

Discriminant

$(1-36z+1458z^2+18225z^3+761076z^4+177147z^5)(4+9z-1539z^2+18954z^3)^2$

Local exponents

≈$-4.272671$ ≈$-0.039841$ ≈$-0.024843$ ≈$-0.024843$ $0$ ≈$0.01303$ ≈$0.01303$ ≈$0.060519-0.040429I$ ≈$0.060519+0.040429I$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $1$ $0$ $1$ $1$ $1$ $1$ $1$
$1$ $3$ $1$ $1$ $0$ $1$ $1$ $3$ $3$ $1$
$2$ $4$ $2$ $2$ $0$ $2$ $2$ $4$ $4$ $1$

≈\(-4.272671\)	≈\(-0.039841\)	≈\(-0.024843\)	≈\(-0.024843\)	\(0\)	≈\(0.01303\)	≈\(0.01303\)	≈\(0.060519-0.040429I\)	≈\(0.060519+0.040429I\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(1\)	\(3\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(3\)	\(3\)	\(1\)
\(2\)	\(4\)	\(2\)	\(2\)	\(0\)	\(2\)	\(2\)	\(4\)	\(4\)	\(1\)

Note:

This is operator "11.19" from ...

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

5

New Number: 8.17 | AESZ: 200 | Superseeker: 19/2 -99607/18 | Hash: e970fa76e74543660fe271b31c8ad485

Degree: 8

$2^{2} \theta^4-2 x\left(106\theta^4+194\theta^3+143\theta^2+46\theta+6\right)-3 x^{2}\left(5305\theta^4-69530\theta^3-87869\theta^2-37122\theta-6174\right)+3^{2} x^{3}\left(406192\theta^4+875286\theta^3+939461\theta^2+616896\theta+144378\right)-3^{6} x^{4}\left(291161\theta^4+1676464\theta^3-1141623\theta^2-986711\theta-230461\right)-3^{10} x^{5}\left(370857\theta^4+845784\theta^3+696122\theta^2+189001\theta+6158\right)+3^{14} x^{6}\left(208867\theta^4+45396\theta^3+18834\theta^2+35097\theta+13814\right)+3^{18} 47 x^{7}\left(2489\theta^4+4972\theta^3+4025\theta^2+1539\theta+232\right)-3^{22} 47^{2} x^{8}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 3, -243, -15315, -55971, ...
--> OEIS
Normalized instanton numbers (n₀=1): 19/2, -5195/8, -99607/18, -217650, 23603349/2, ... ; Common denominator:...

Discriminant

$-(531441z^4-347733z^3-7290z^2+50z-1)(-2+3z+11421z^2)^2$

Local exponents

$-\frac{ 1}{ 7614}-\frac{ 1}{ 7614}\sqrt{ 10153}$ $0$ $-\frac{ 1}{ 7614}+\frac{ 1}{ 7614}\sqrt{ 10153}$ $#ND+#NDI$ $\infty$
$0$ $0$ $0$ $0$ $1$
$1$ $0$ $1$ $1$ $1$
$3$ $0$ $3$ $1$ $1$
$4$ $0$ $4$ $2$ $1$

\(-\frac{ 1}{ 7614}-\frac{ 1}{ 7614}\sqrt{ 10153}\)	\(0\)	\(-\frac{ 1}{ 7614}+\frac{ 1}{ 7614}\sqrt{ 10153}\)	\(#ND+#NDI\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(3\)	\(0\)	\(3\)	\(1\)	\(1\)
\(4\)	\(0\)	\(4\)	\(2\)	\(1\)

Note:

This operator has a second MUM-point at infinity, corresponding to operator 8.16

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

6

New Number: 9.3 | AESZ: | Superseeker: 10 3394/3 | Hash: 40e3715abcc5c4cb07e700ca79f80abf

Degree: 9

$\theta^4-x\left(57\theta^4+116\theta^3+84\theta^2+26\theta+3\right)-2 x^{2}\left(894\theta^4+3208\theta^3+4571\theta^2+2771\theta+651\right)-2 x^{3}\left(7322\theta^4+56368\theta^3+124783\theta^2+101099\theta+29757\right)+2 3^{2} x^{4}\left(6967\theta^4-27080\theta^3-139991\theta^2-138507\theta-45297\right)+2 3^{4} x^{5}\left(17617\theta^4+49068\theta^3-31255\theta^2-79893\theta-34578\right)+2 3^{8} x^{6}\left(1082\theta^4+8360\theta^3+7967\theta^2+1439\theta-773\right)-2 3^{11} x^{7}\left(198\theta^4-864\theta^3-1545\theta^2-909\theta-155\right)-3^{15} x^{8}\left(69\theta^4+144\theta^3+126\theta^2+54\theta+10\right)-3^{20} x^{9}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 3, 135, 5349, 258039, ...
--> OEIS
Normalized instanton numbers (n₀=1): 10, 77, 3394/3, 24029, 640402, ... ; Common denominator:...

Discriminant

$-(-1+81z)(-1+9z)^2(81z^2+14z+1)^3$

Local exponents

$-\frac{ 7}{ 81}-\frac{ 4}{ 81}\sqrt{ 2}I$ $-\frac{ 7}{ 81}+\frac{ 4}{ 81}\sqrt{ 2}I$ $0$ $\frac{ 1}{ 81}$ $\frac{ 1}{ 9}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $1$
$\frac{ 1}{ 2}$ $\frac{ 1}{ 2}$ $0$ $1$ $1$ $1$
$\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$ $0$ $1$ $3$ $1$
$2$ $2$ $0$ $2$ $4$ $1$

\(-\frac{ 7}{ 81}-\frac{ 4}{ 81}\sqrt{ 2}I\)	\(-\frac{ 7}{ 81}+\frac{ 4}{ 81}\sqrt{ 2}I\)	\(0\)	\(\frac{ 1}{ 81}\)	\(\frac{ 1}{ 9}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(\frac{ 1}{ 2}\)	\(\frac{ 1}{ 2}\)	\(0\)	\(1\)	\(1\)	\(1\)
\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)	\(0\)	\(1\)	\(3\)	\(1\)
\(2\)	\(2\)	\(0\)	\(2\)	\(4\)	\(1\)

Note:

This is operator "9.3" from ...

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:

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