Calabi-Yau differential operator database v.3.0

1

New Number: 2.12 | AESZ: 64 | Superseeker: 432 78259376 | Hash: 43991f21e20c16ab91690259b788b4cd

Degree: 2

$\theta^4-2^{2} 3 x(6\theta+1)(6\theta+5)(10\theta^2+10\theta+3)+2^{4} 3^{4} x^{2}(6\theta+1)(6\theta+5)(6\theta+7)(6\theta+11)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 180, 207900, 379819440, 855338063580, ...
--> OEIS
Normalized instanton numbers (n₀=1): 432, 130842, 78259376, 68104755558, 73096116588720, ... ; Common denominator:...

Discriminant

$(3888z-1)(432z-1)$

Local exponents

$0$ $\frac{ 1}{ 3888}$ $\frac{ 1}{ 432}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 6}$
$0$ $1$ $1$ $\frac{ 5}{ 6}$
$0$ $1$ $1$ $\frac{ 7}{ 6}$
$0$ $2$ $2$ $\frac{ 11}{ 6}$

\(0\)	\(\frac{ 1}{ 3888}\)	\(\frac{ 1}{ 432}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 6}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 5}{ 6}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 7}{ 6}\)
\(0\)	\(2\)	\(2\)	\(\frac{ 11}{ 6}\)

Note:

Hadamard product $B\ast c$.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-364\)	\(72\)	\(6\)	\(7\)

2

New Number: 2.4 | AESZ: 62 | Superseeker: 372 71562236 | Hash: 07a3fd7577f878056e765831c6820f3d

Degree: 2

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 120, 138600, 228708480, 463140798120, ...
--> OEIS
Normalized instanton numbers (n₀=1): 372, 136182, 71562236, 63364481358, 65860679690400, ... ; Common denominator:...

Discriminant

$-(432z+1)(3456z-1)$

Local exponents

$-\frac{ 1}{ 432}$ $0$ $\frac{ 1}{ 3456}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 6}$
$1$ $0$ $1$ $\frac{ 5}{ 6}$
$1$ $0$ $1$ $\frac{ 7}{ 6}$
$2$ $0$ $2$ $\frac{ 11}{ 6}$

\(-\frac{ 1}{ 432}\)	\(0\)	\(\frac{ 1}{ 3456}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 6}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 5}{ 6}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 7}{ 6}\)
\(2\)	\(0\)	\(2\)	\(\frac{ 11}{ 6}\)

Note:

Hadamard product D*a

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-366\)	\(72\)	\(6\)	\(7\)

3

New Number: 3.13 | AESZ: | Superseeker: 352 15001120/3 | Hash: b5a6f76d274395537de2c3169fdac9bf

Degree: 3

$\theta^4-2^{2} x\left(688\theta^4+1232\theta^3+902\theta^2+286\theta+33\right)+2^{4} 3^{2} x^{2}(4\theta+3)(3776\theta^3+10096\theta^2+8268\theta+1515)-2^{10} 3^{4} 5^{2} x^{3}(4\theta+1)(4\theta+3)(4\theta+7)(4\theta+9)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 132, 62748, 43686384, 37830871260, ...
--> OEIS
Normalized instanton numbers (n₀=1): 352, 18676, 15001120/3, 1489325052, 586526654304, ... ; Common denominator:...

Discriminant

$-(1600z-1)(-1+576z)^2$

Local exponents

$0$ $\frac{ 1}{ 1600}$ $\frac{ 1}{ 576}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 4}$
$0$ $1$ $\frac{ 1}{ 2}$ $\frac{ 3}{ 4}$
$0$ $1$ $1$ $\frac{ 7}{ 4}$
$0$ $2$ $\frac{ 3}{ 2}$ $\frac{ 9}{ 4}$

\(0\)	\(\frac{ 1}{ 1600}\)	\(\frac{ 1}{ 576}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 4}\)
\(0\)	\(1\)	\(\frac{ 1}{ 2}\)	\(\frac{ 3}{ 4}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 7}{ 4}\)
\(0\)	\(2\)	\(\frac{ 3}{ 2}\)	\(\frac{ 9}{ 4}\)

Note:

Operator equivalent to AESZ 229

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-56\)	\(36\)	\(6\)	\(4\)

4

New Number: 3.18 | AESZ: 388 | Superseeker: 266 11433160/3 | Hash: 7e11db69c1b7bd8781e54a5eadb0e307

Degree: 3

$\theta^4-2 x\left(582\theta^4+1164\theta^3+815\theta^2+233\theta+25\right)+2^{2} x^{2}(\theta+1)^2(2316\theta^2+4632\theta+1907)-2^{4} 17^{2} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 50, 17142, 9383540, 6301530550, ...
--> OEIS
Normalized instanton numbers (n₀=1): 266, 19320, 11433160/3, 1106069392, 397606861972, ... ; Common denominator:...

Discriminant

$-(1156z-1)(4z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 1156}$ $\frac{ 1}{ 4}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $0$ $1$
$0$ $1$ $1$ $2$
$0$ $2$ $1$ $\frac{ 5}{ 2}$

\(0\)	\(\frac{ 1}{ 1156}\)	\(\frac{ 1}{ 4}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(0\)	\(1\)
\(0\)	\(1\)	\(1\)	\(2\)
\(0\)	\(2\)	\(1\)	\(\frac{ 5}{ 2}\)

Note:

This is operator "3.18" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-64\)	\(36\)	\(6\)	\(4\)

5

New Number: 3.6 | AESZ: ~33 | Superseeker: 196 2993772 | Hash: 29aeacb8c7e91c8c2838e65ce2750b5a

Degree: 3

$\theta^4+2^{2} x\left(60\theta^4-8\theta^3+31\theta^2+35\theta+6\right)-2^{10} x^{2}(4\theta+3)(132\theta^3+395\theta^2+363\theta+69)-2^{14} 7^{2} x^{3}(4\theta+1)(4\theta+3)(4\theta+7)(4\theta+9)$

Maple LaTex

Coefficients of the holomorphic solution: 1, -24, 13992, -920832, 1808021160, ...
--> OEIS
Normalized instanton numbers (n₀=1): 196, 17212, 2993772, 789858520, 260782261024, ... ; Common denominator:...

Discriminant

$-(784z-1)(1+512z)^2$

Local exponents

$-\frac{ 1}{ 512}$ $0$ $\frac{ 1}{ 784}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 4}$
$\frac{ 1}{ 2}$ $0$ $1$ $\frac{ 3}{ 4}$
$1$ $0$ $1$ $\frac{ 7}{ 4}$
$\frac{ 3}{ 2}$ $0$ $2$ $\frac{ 9}{ 4}$

\(-\frac{ 1}{ 512}\)	\(0\)	\(\frac{ 1}{ 784}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 4}\)
\(\frac{ 1}{ 2}\)	\(0\)	\(1\)	\(\frac{ 3}{ 4}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 7}{ 4}\)
\(\frac{ 3}{ 2}\)	\(0\)	\(2\)	\(\frac{ 9}{ 4}\)

Note:

Operator AESZ 33 is replaced by this equivalent operator.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-72\)	\(36\)	\(6\)	\(4\)

6

New Number: 1.11 | AESZ: 11 | Superseeker: 324 10792428 | Hash: 8ac8b98b80383c9f0ea125ccd6e6a55d

Degree: 1

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 72, 37800, 31046400, 31216185000, ...
--> OEIS
Normalized instanton numbers (n₀=1): 324, 37260, 10792428, 4580482284, 2405245303584, ... ; Common denominator:...

Discriminant

$1-1728z$

Local exponents

$0$ $\frac{ 1}{ 1728}$ $\infty$
$0$ $0$ $\frac{ 1}{ 4}$
$0$ $1$ $\frac{ 1}{ 3}$
$0$ $1$ $\frac{ 2}{ 3}$
$0$ $2$ $\frac{ 3}{ 4}$

\(0\)	\(\frac{ 1}{ 1728}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 4}\)
\(0\)	\(1\)	\(\frac{ 1}{ 3}\)
\(0\)	\(1\)	\(\frac{ 2}{ 3}\)
\(0\)	\(2\)	\(\frac{ 3}{ 4}\)

Note:

A-incarnation: X(4,6) in P^5(1,1,1,2,2,3)

\(4.+5.291856516I\)	\(-1-364\lambda\)	\(\frac{ 3}{ 2}+546\lambda\)	\(1.055763632-364\lambda\)
\(9\)	\(-2\)	\(\frac{ 9}{ 2}\)	\(-\frac{ 3}{ 2}-546\lambda\)
\(6\)	\(-2\)	\(4\)	\(-1-364\lambda\)
\(18\)	\(-6\)	\(9\)	\(-2.-5.291856516I\)

\(4.+3.547288434I\)	\(-\frac{ 3}{ 2}-366\lambda\)	\(1.7500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000+2.069251586I\)	\(.298604603-366\lambda\)
\(7\)	\(-\frac{ 5}{ 2}\)	\(\frac{ 49}{ 12}\)	\(-1.7500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000-2.069251586I\)
\(6\)	\(-3\)	\(\frac{ 9}{ 2}\)	\(-\frac{ 3}{ 2}-366\lambda\)
\(12\)	\(-6\)	\(7\)	\(-2.-3.547288434I\)

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

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

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

\(1+364\lambda\)	\(0\)	\(182\lambda\)	\(.518587877\)
\(3\)	\(1\)	\(\frac{ 3}{ 2}\)	\(-182\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(6\)	\(0\)	\(3\)	\(1-364\lambda\)

\(-364\lambda\)	\(4\)	\(1\)	\(1\)
\(-3\)	\(-3\)	\(-1\)	\(0\)
\(0\)	\(6\)	\(0\)	\(0\)
\(-6\)	\(0\)	\(0\)	\(0\)

\(1+366\lambda\)	\(0\)	\(183\lambda\)	\(.524302301\)
\(3\)	\(1\)	\(\frac{ 3}{ 2}\)	\(-183\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(6\)	\(0\)	\(3\)	\(1-366\lambda\)

\(-366\lambda\)	\(4\)	\(1\)	\(1\)
\(-3\)	\(-3\)	\(-1\)	\(0\)
\(0\)	\(6\)	\(0\)	\(0\)
\(-6\)	\(0\)	\(0\)	\(0\)

\(-56\lambda\)	\(\frac{ 5}{ 2}\)	\(1\)	\(1\)
\(-\frac{ 3}{ 2}\)	\(-3\)	\(-1\)	\(0\)
\(0\)	\(6\)	\(0\)	\(0\)
\(-6\)	\(0\)	\(0\)	\(0\)

\(-64\lambda\)	\(\frac{ 5}{ 2}\)	\(1\)	\(1\)
\(-\frac{ 3}{ 2}\)	\(-3\)	\(-1\)	\(0\)
\(0\)	\(6\)	\(0\)	\(0\)
\(-6\)	\(0\)	\(0\)	\(0\)