Calabi-Yau differential operator database v.3.0

1

New Number: 2.27 | AESZ: 140 | Superseeker: 108 -4945756 | Hash: 74692097f8183c067c2c4b1a5c93387b

Degree: 2

$\theta^4-2^{2} 3 x(6\theta+1)(6\theta+5)(17\theta^2+17\theta+6)+2^{7} 3^{4} x^{2}(6\theta+1)(6\theta+5)(6\theta+7)(6\theta+11)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 360, 582120, 1274232960, 3204505984680, ...
--> OEIS
Normalized instanton numbers (n₀=1): 108, 54135, -4945756, 7925523138, -2434666062240, ... ; Common denominator:...

Discriminant

$(3888z-1)(3456z-1)$

Local exponents

$0$ $\frac{ 1}{ 3888}$ $\frac{ 1}{ 3456}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 6}$
$0$ $1$ $1$ $\frac{ 5}{ 6}$
$0$ $1$ $1$ $\frac{ 7}{ 6}$
$0$ $2$ $2$ $\frac{ 11}{ 6}$

\(0\)	\(\frac{ 1}{ 3888}\)	\(\frac{ 1}{ 3456}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 6}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 5}{ 6}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 7}{ 6}\)
\(0\)	\(2\)	\(2\)	\(\frac{ 11}{ 6}\)

Note:

Hadamard product $D \ast g$

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-676\)	\(132\)	\(12\)	\(13\)

2

New Number: 2.64 | AESZ: 182 | Superseeker: 1 7 | Hash: 89ba4729efa82413b33fe6928ff8d2c9

Degree: 2

$\theta^4-x\left(43\theta^4+86\theta^3+77\theta^2+34\theta+6\right)+2^{2} 3 x^{2}(\theta+1)^2(6\theta+5)(6\theta+7)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 6, 66, 924, 14850, ...
--> OEIS
Normalized instanton numbers (n₀=1): 1, 7/4, 7, 40, 270, ... ; Common denominator:...

Discriminant

$(27z-1)(16z-1)$

Local exponents

$0$ $\frac{ 1}{ 27}$ $\frac{ 1}{ 16}$ $\infty$
$0$ $0$ $0$ $\frac{ 5}{ 6}$
$0$ $1$ $1$ $1$
$0$ $1$ $1$ $1$
$0$ $2$ $2$ $\frac{ 7}{ 6}$

\(0\)	\(\frac{ 1}{ 27}\)	\(\frac{ 1}{ 16}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 5}{ 6}\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(2\)	\(2\)	\(\frac{ 7}{ 6}\)

Note:

This is operator "2.64" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-96\)	\(132\)	\(132\)	\(33\)

3

New Number: 5.25 | AESZ: 198 | Superseeker: -84/11 -9052/11 | Hash: a1f924763b047c2720d99cfca5ca63db

Degree: 5

$11^{2} \theta^4+7 11 x\left(130\theta^4+266\theta^3+210\theta^2+77\theta+11\right)-x^{2}\left(11198+55253\theta+103725\theta^2+89990\theta^3+32126\theta^4\right)+x^{3}\left(1716+20625\theta+63474\theta^2+74184\theta^3+28723\theta^4\right)-7 x^{4}\left(1135\theta^4+2336\theta^3+1881\theta^2+713\theta+110\right)+7^{2} x^{5}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, -7, 199, -8359, 423751, ...
--> OEIS
Normalized instanton numbers (n₀=1): -84/11, 639/11, -9052/11, 189021/11, -4838013/11, ... ; Common denominator:...

Discriminant

$(z^3-159z^2+84z+1)(-11+7z)^2$

Local exponents

≈$-0.011648$ $0$ ≈$0.541757$ $\frac{ 11}{ 7}$ ≈$158.469891$ $\infty$
$0$ $0$ $0$ $0$ $0$ $1$
$1$ $0$ $1$ $1$ $1$ $1$
$1$ $0$ $1$ $3$ $1$ $1$
$2$ $0$ $2$ $4$ $2$ $1$

≈\(-0.011648\)	\(0\)	≈\(0.541757\)	\(\frac{ 11}{ 7}\)	≈\(158.469891\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)
\(1\)	\(0\)	\(1\)	\(3\)	\(1\)	\(1\)
\(2\)	\(0\)	\(2\)	\(4\)	\(2\)	\(1\)

Note:

There is a second MUM-point at infinity, corresponding to Operator AESZ 193/5.22

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-408\)	\(132\)	\(132\)	\(33\)

4

New Number: 6.9 | AESZ: | Superseeker: 31/81 29/9 | Hash: 98af5121f39c27098356e3ade277f975

Degree: 6

$3^{8} \theta^4-3^{4} x\left(1234\theta^4+2168\theta^3+1975\theta^2+891\theta+162\right)-x^{2}\left(428004+1521180\theta+2033921\theta^2+1177556\theta^3+205589\theta^4\right)+x^{3}\left(2310517\theta^4+12882402\theta^3+26939429\theta^2+25052328\theta+8683524\right)-2^{2} 5^{2} x^{4}\left(51526\theta^4+332687\theta^3+804453\theta^2+849398\theta+325796\right)+2^{2} 5^{4} x^{5}(\theta+1)(1593\theta^3+8667\theta^2+15104\theta+8516)-2^{4} 5^{6} x^{6}(\theta+2)(\theta+1)(2\theta+3)^2$

Maple LaTex

Coefficients of the holomorphic solution: 1, 2, 14, 104, 1030, ...
--> OEIS
Normalized instanton numbers (n₀=1): 31/81, 40/27, 29/9, 1532/81, 6551/81, ... ; Common denominator:...

Discriminant

$-(16z-1)(25z^3-17z^2+2z+1)(-81+50z)^2$

Local exponents

≈$-0.17455$ $0$ $\frac{ 1}{ 16}$ ≈$0.427275-0.215865I$ ≈$0.427275+0.215865I$ $\frac{ 81}{ 50}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $0$ $1$ $1$ $1$ $1$ $\frac{ 3}{ 2}$
$1$ $0$ $1$ $1$ $1$ $3$ $\frac{ 3}{ 2}$
$2$ $0$ $2$ $2$ $2$ $4$ $2$

≈\(-0.17455\)	\(0\)	\(\frac{ 1}{ 16}\)	≈\(0.427275-0.215865I\)	≈\(0.427275+0.215865I\)	\(\frac{ 81}{ 50}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 3}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(3\)	\(\frac{ 3}{ 2}\)
\(2\)	\(0\)	\(2\)	\(2\)	\(2\)	\(4\)	\(2\)

Note:

This is operator "6.9" from ...

\(-5.088262696-2.403669772I\)	\(-9.231233148+8.267057761I\)	\(6.171205313+7.544982974I\)	\(3.227450678-3.068539845I\)
\(-6.270037138+8.225054977I\)	\(9.605336680+17.587574161I\)	\(14.123297626-6.142790642I\)	\(-3.279565479-6.225556868I\)
\(-2.195545276+2.457029703I\)	\(2.297135143+5.799775915I\)	\(5.638341607-1.600871223I\)	\(-.887292619-2.058764958I\)
\(-13.680081028+17.945574495I\)	\(18.775280028+38.372889078I\)	\(30.814467548-13.402452309I\)	\(-6.155415591-13.583033166I\)

\(\frac{ 17}{ 4}+288\lambda\)	\(-\frac{ 13}{ 24}-48\lambda\)	\(\frac{ 13}{ 96}+12\lambda\)	\(-.21754163e-1-\frac{ 52}{ 11}\lambda\)
\(\frac{ 33}{ 2}\)	\(-\frac{ 7}{ 4}\)	\(\frac{ 11}{ 16}\)	\(-\frac{ 13}{ 96}-12\lambda\)
\(66\)	\(-11\)	\(\frac{ 15}{ 4}\)	\(-\frac{ 13}{ 24}-48\lambda\)
\(396\)	\(-66\)	\(\frac{ 33}{ 2}\)	\(-\frac{ 9}{ 4}-288\lambda\)

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

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

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

\(1.+3.275911176I\)	\(0\)	\(1.501459289I\)	\(.894299503\)
\(\frac{ 11}{ 2}\)	\(1\)	\(\frac{ 121}{ 48}\)	\(-1.501459289I\)
\(0\)	\(0\)	\(1\)	\(0\)
\(12\)	\(0\)	\(\frac{ 11}{ 2}\)	\(1.-3.275911176I\)

\(-\frac{ 409488897}{ 125000000}I\)	\(\frac{ 15}{ 2}\)	\(1\)	\(1\)
\(-\frac{ 11}{ 2}\)	\(-6\)	\(-1\)	\(0\)
\(0\)	\(12\)	\(0\)	\(0\)
\(-12\)	\(0\)	\(0\)	\(0\)

\(1+96\lambda\)	\(0\)	\(4\lambda\)	\(.1639606e-2\)
\(\frac{ 11}{ 2}\)	\(1\)	\(\frac{ 11}{ 48}\)	\(-4\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(132\)	\(0\)	\(\frac{ 11}{ 2}\)	\(1-96\lambda\)

\(-96\lambda\)	\(\frac{ 55}{ 2}\)	\(1\)	\(1\)
\(-\frac{ 11}{ 2}\)	\(-66\)	\(-1\)	\(0\)
\(0\)	\(132\)	\(0\)	\(0\)
\(-132\)	\(0\)	\(0\)	\(0\)