Calabi-Yau differential operator database v.3.0

1

New Number: 2.10 | AESZ: 70 | Superseeker: 27 18089 | Hash: 3d2adae6eaf26a56c76b8b67d92cc5df

Degree: 2

$\theta^4-3 x(3\theta+1)(3\theta+2)(10\theta^2+10\theta+3)+3^{4} x^{2}(3\theta+1)(3\theta+2)(3\theta+4)(3\theta+5)$

Coefficients of the holomorphic solution: 1, 18, 1350, 156240, 22141350, ...
--> OEIS
Normalized instanton numbers (n₀=1): 27, 432, 18089, 997785, 68438142, ... ; Common denominator:...

Discriminant

$(243z-1)(27z-1)$

Local exponents

$0$ $\frac{ 1}{ 243}$ $\frac{ 1}{ 27}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 3}$
$0$ $1$ $1$ $\frac{ 2}{ 3}$
$0$ $1$ $1$ $\frac{ 4}{ 3}$
$0$ $2$ $2$ $\frac{ 5}{ 3}$

\(0\)	\(\frac{ 1}{ 243}\)	\(\frac{ 1}{ 27}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 2}{ 3}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 4}{ 3}\)
\(0\)	\(2\)	\(2\)	\(\frac{ 5}{ 3}\)

Note:

Hadamard product $B\ast c$.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-156\)	\(72\)	\(18\)	\(9\)

2

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	,...

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

3

New Number: 3.31 | AESZ: | Superseeker: 4 284 | Hash: 660b0951ad934fc17fda7eb9b1750649

Degree: 3

$\theta^4-2^{2} x(2\theta+1)^2(5\theta^2+5\theta+2)+2^{5} x^{2}(2\theta+1)(2\theta+3)(7\theta^2+14\theta+8)-2^{4} 11 x^{3}(2\theta+1)(2\theta+3)^2(2\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 168, 5360, 210280, ...
--> OEIS
Normalized instanton numbers (n₀=1): 4, 27, 284, 4368, 80968, ... ; Common denominator:...

Discriminant

$1-80z+896z^2-2816z^3$

No data for singularities

Note:

This is operator "3.31" from ...

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

4

New Number: 4.37 | AESZ: 206 | Superseeker: 4 284 | Hash: bd5dae321e1369e7fae153775f84a351

Degree: 4

$\theta^4-2^{2} x\theta(\theta+1)(2\theta+1)^2-2^{5} x^{2}(2\theta+1)(2\theta+3)(11\theta^2+22\theta+12)-2^{4} 3 5^{2} x^{3}(2\theta+1)(2\theta+3)^2(2\theta+5)-2^{8} 19 x^{4}(2\theta+1)(2\theta+3)(2\theta+5)(2\theta+7)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 72, 1200, 44520, ...
--> OEIS
Normalized instanton numbers (n₀=1): 4, 27, 284, 4368, 80968, ... ; Common denominator:...

Discriminant

$-(16z+1)(4864z^3+896z^2+32z-1)$

Local exponents

≈$-0.10185-0.013248I$ ≈$-0.10185+0.013248I$ $-\frac{ 1}{ 16}$ $0$ $s_1$ $s_3$ $s_2$ ≈$0.019489$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $1$ $1$ $0$ $1$ $1$ $1$ $1$ $\frac{ 3}{ 2}$
$1$ $1$ $1$ $0$ $1$ $1$ $1$ $1$ $\frac{ 5}{ 2}$
$2$ $2$ $2$ $0$ $2$ $2$ $2$ $2$ $\frac{ 7}{ 2}$

≈\(-0.10185-0.013248I\)	≈\(-0.10185+0.013248I\)	\(-\frac{ 1}{ 16}\)	\(0\)	\(s_1\)	\(s_3\)	\(s_2\)	≈\(0.019489\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 3}{ 2}\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 5}{ 2}\)
\(2\)	\(2\)	\(2\)	\(0\)	\(2\)	\(2\)	\(2\)	\(2\)	\(\frac{ 7}{ 2}\)

Note:

Sporadic Operator.

\(4+468\lambda\)	\(-1.-.755979502I\)	\(\frac{ 1}{ 2}+78\lambda\)	\(-.71415832e-1-52\lambda\)
\(9\)	\(-2\)	\(\frac{ 3}{ 2}\)	\(-\frac{ 1}{ 2}-78\lambda\)
\(18\)	\(-6\)	\(4\)	\(-1.-.755979502I\)
\(54\)	\(-18\)	\(9\)	\(-2-468\lambda\)

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

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:

Explicit solution

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

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

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

\(1.+.755979502I\)	\(0\)	\(26\lambda\)	\(.31750278e-1\)
\(3\)	\(1\)	\(\frac{ 1}{ 2}\)	\(-26\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(18\)	\(0\)	\(3\)	\(1.-.755979502I\)

\(-\frac{ 377989751}{ 500000000}I\)	\(6\)	\(1\)	\(1\)
\(-3\)	\(-9\)	\(-1\)	\(0\)
\(0\)	\(18\)	\(0\)	\(0\)
\(-18\)	\(0\)	\(0\)	\(0\)

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

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