Calabi-Yau differential operator database v.3.0

1

New Number: 2.24 | AESZ: 137 | Superseeker: 20 1684/3 | Hash: 198d6c822d6c46225ac2553d60df6539

Degree: 2

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

Coefficients of the holomorphic solution: 1, 24, 1512, 124800, 11730600, ...
--> OEIS
Normalized instanton numbers (n₀=1): 20, 2, 1684/3, 7602, 173472, ... ; Common denominator:...

Discriminant

$(144z-1)(128z-1)$

Local exponents

$0$ $\frac{ 1}{ 144}$ $\frac{ 1}{ 128}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $\frac{ 3}{ 2}$
$0$ $2$ $2$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 1}{ 144}\)	\(\frac{ 1}{ 128}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 3}{ 2}\)
\(0\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

Hadamard product $A \ast g$.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-16\)	\(96\)	\(48\)	\(16\)

2

New Number: 2.52 | AESZ: 16 | Superseeker: 4 644/3 | Hash: 05af0662662bfbec63e3186c4f363313

Degree: 2

$\theta^4-2^{2} x(2\theta+1)^2(5\theta^2+5\theta+2)+2^{8} x^{2}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 168, 5120, 190120, ...
--> OEIS
Normalized instanton numbers (n₀=1): 4, 20, 644/3, 3072, 52512, ... ; Common denominator:...

Discriminant

$(64z-1)(16z-1)$

Local exponents

$0$ $\frac{ 1}{ 64}$ $\frac{ 1}{ 16}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $1$
$0$ $1$ $1$ $1$
$0$ $2$ $2$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 1}{ 64}\)	\(\frac{ 1}{ 16}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

Hadamard product $I \ast \alpha$
A-Incarnation: diagonal subfamily of 1,1,1,1-intersection in $P^1 \times P^1 \times P^1 \times \P^1$
B-Incarnations:
Fibre products: 62211- x 632--1, S62211

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-128\)	\(96\)	\(48\)	\(16\)

3

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	,...

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

4

New Number: 8.10 | AESZ: 123 | Superseeker: 12 1828/3 | Hash: f0d76ab2b6b8808f4faa4ab8ecadff2c

Degree: 8

$\theta^4-2^{2} x(3\theta^2+3\theta+1)(10\theta^2+10\theta+3)+2^{4} x^{2}\left(209\theta^4+1052\theta^3+1471\theta^2+838\theta+183\right)+2^{7} 3^{2} x^{3}\left(30\theta^4-180\theta^3-551\theta^2-417\theta-111\right)-2^{10} 3^{2} x^{4}\left(227\theta^4+454\theta^3-550\theta^2-777\theta-261\right)+2^{12} 3^{4} x^{5}\left(30\theta^4+300\theta^3+169\theta^2-25\theta-35\right)+2^{14} 3^{4} x^{6}\left(209\theta^4-216\theta^3-431\theta^2-216\theta-27\right)-2^{17} 3^{6} x^{7}(3\theta^2+3\theta+1)(10\theta^2+10\theta+3)+2^{20} 3^{8} x^{8}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 300, 10416, 431964, ...
--> OEIS
Normalized instanton numbers (n₀=1): 12, -47/2, 1828/3, -10813/4, 127948, ... ; Common denominator:...

Discriminant

$(36z-1)(8z-1)(72z-1)(4z-1)(-1+288z^2)^2$

Local exponents

$-\frac{ 1}{ 24}\sqrt{ 2}$ $0$ $\frac{ 1}{ 72}$ $\frac{ 1}{ 36}$ $\frac{ 1}{ 24}\sqrt{ 2}$ $\frac{ 1}{ 8}$ $\frac{ 1}{ 4}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $0$ $1$ $1$ $1$ $1$ $1$ $1$
$3$ $0$ $1$ $1$ $3$ $1$ $1$ $1$
$4$ $0$ $2$ $2$ $4$ $2$ $2$ $1$

\(-\frac{ 1}{ 24}\sqrt{ 2}\)	\(0\)	\(\frac{ 1}{ 72}\)	\(\frac{ 1}{ 36}\)	\(\frac{ 1}{ 24}\sqrt{ 2}\)	\(\frac{ 1}{ 8}\)	\(\frac{ 1}{ 4}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(3\)	\(0\)	\(1\)	\(1\)	\(3\)	\(1\)	\(1\)	\(1\)
\(4\)	\(0\)	\(2\)	\(2\)	\(4\)	\(2\)	\(2\)	\(1\)

Note:

Hadamard product $c \ast d$. 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 $\Q(\sqrt{?})$.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(28\)	\(72\)	\(48\)	\(14\)

5

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 ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-86\)	\(84\)	\(48\)	\(15\)

6

New Number: 8.3 | AESZ: 105 | Superseeker: 8 -104 | Hash: 7b27135451cf2016217211c633b7ab83

Degree: 8

$\theta^4-2^{2} x(3\theta^2+3\theta+1)(7\theta^2+7\theta+2)+2^{5} 3 x^{2}\left(15\theta^4+28\theta^3+39\theta^2+22\theta+4\right)-2^{10} x^{3}\left(21\theta^4-126\theta^3-386\theta^2-291\theta-76\right)+2^{14} x^{4}\left(37\theta^4+74\theta^3+50\theta^2+13\theta+6\right)+2^{18} x^{5}\left(21\theta^4+210\theta^3+118\theta^2-19\theta-24\right)+2^{21} 3 x^{6}\left(15\theta^4+32\theta^3+45\theta^2+32\theta+8\right)+2^{26} x^{7}(3\theta^2+3\theta+1)(7\theta^2+7\theta+2)+2^{32} x^{8}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 200, 6272, 233896, ...
--> OEIS
Normalized instanton numbers (n₀=1): 8, 71/2, -104, 4202, 50112, ... ; Common denominator:...

Discriminant

$(8z+1)(64z-1)(4z+1)(32z-1)(1+256z^2)^2$

Local exponents

$-\frac{ 1}{ 4}$ $-\frac{ 1}{ 8}$ $0-\frac{ 1}{ 16}I$ $0$ $0+\frac{ 1}{ 16}I$ $\frac{ 1}{ 64}$ $\frac{ 1}{ 32}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $0$ $1$ $1$ $1$ $1$
$1$ $1$ $3$ $0$ $3$ $1$ $1$ $1$
$2$ $2$ $4$ $0$ $4$ $2$ $2$ $1$

\(-\frac{ 1}{ 4}\)	\(-\frac{ 1}{ 8}\)	\(0-\frac{ 1}{ 16}I\)	\(0\)	\(0+\frac{ 1}{ 16}I\)	\(\frac{ 1}{ 64}\)	\(\frac{ 1}{ 32}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(3\)	\(0\)	\(3\)	\(1\)	\(1\)	\(1\)
\(2\)	\(2\)	\(4\)	\(0\)	\(4\)	\(2\)	\(2\)	\(1\)

Note:

Hadamard product $a \ast d$. 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
$Q(\sqrt{-1})$.

\(.924152876+.943952044I\)	\(.188111204+1.275393351I\)	\(.593885779-.164736576I\)	\(-.98788356e-1-.111719004I\)
\(.205812544+3.214663066I\)	\(2.256285526+4.201440782I\)	\(1.918711704-.844672672I\)	\(-.387126151-.327818207I\)
\(-.81202836e-1+5.639775328I\)	\(1.617313859+7.506248400I\)	\(4.465075410-1.211574339I\)	\(-.630856884-.62528948587308939323761000463177396943029180176007410838351088466882816118573413617415470125057897174617878647522000926354793886058360352014821676702176933765632237146827234830940250115794349235757295I\)
\(2.469750531+38.575956789I\)	\(15.075426310+50.417289384I\)	\(23.024540442-10.136072063I\)	\(-3.645513812-3.933818487I\)

\(5+512\lambda\)	\(-1.-.620290874I\)	\(\frac{ 1}{ 3}+\frac{ 128}{ 3}\lambda\)	\(-\frac{ 1}{ 7}7^{ \frac{ 5}{ 8})exp(-\frac{ 9}{ 4}}Zeta(5)^(\frac{ 1}{ 4})-\frac{ 64}{ 3}\lambda\)
\(16\)	\(-3\)	\(\frac{ 4}{ 3}\)	\(-\frac{ 1}{ 3}-\frac{ 128}{ 3}\lambda\)
\(48\)	\(-12\)	\(5\)	\(-1.-.620290874I\)
\(192\)	\(-48\)	\(16\)	\(-3-512\lambda\)

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

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

\(1+16\lambda\)	\(0\)	\(\frac{ 4}{ 3}\lambda\)	\(.125248e-3\)
\(4\)	\(1\)	\(\frac{ 1}{ 3}\)	\(-\frac{ 4}{ 3}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(48\)	\(0\)	\(4\)	\(1-16\lambda\)

\(-16\lambda\)	\(12\)	\(1\)	\(1\)
\(-4\)	\(-24\)	\(-1\)	\(0\)
\(0\)	\(48\)	\(0\)	\(0\)
\(-48\)	\(0\)	\(0\)	\(0\)

\(1.+.620290874I\)	\(0\)	\(\frac{ 32}{ 3}\lambda\)	\(.8015849e-2\)
\(4\)	\(1\)	\(\frac{ 1}{ 3}\)	\(-\frac{ 32}{ 3}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(48\)	\(0\)	\(4\)	\(1.-.620290874I\)

\(-\frac{ 310145437}{ 500000000}I\)	\(12\)	\(1\)	\(1\)
\(-4\)	\(-24\)	\(-1\)	\(0\)
\(0\)	\(48\)	\(0\)	\(0\)
\(-48\)	\(0\)	\(0\)	\(0\)