Calabi-Yau differential operator database v.3.0

1

New Number: 2.29 | AESZ: 142 | Superseeker: 45 27735 | Hash: 85c2c2d8111a859d4cc03e8892c56af9

Degree: 2

$\theta^4-3^{2} x(3\theta+1)(3\theta+2)(18\theta^2+18\theta+7)+3^{8} x^{2}(3\theta+1)(3\theta+2)(3\theta+4)(3\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 126, 44550, 20447280, 10600093350, ...
--> OEIS
Normalized instanton numbers (n₀=1): 45, -3465/4, 27735, -1156005, 55721970, ... ; Common denominator:...

Discriminant

$(729z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 729}$ $\infty$
$0$ $0$ $\frac{ 1}{ 3}$
$0$ $\frac{ 1}{ 3}$ $\frac{ 2}{ 3}$
$0$ $\frac{ 2}{ 3}$ $\frac{ 4}{ 3}$
$0$ $1$ $\frac{ 5}{ 3}$

\(0\)	\(\frac{ 1}{ 729}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(0\)	\(\frac{ 1}{ 3}\)	\(\frac{ 2}{ 3}\)
\(0\)	\(\frac{ 2}{ 3}\)	\(\frac{ 4}{ 3}\)
\(0\)	\(1\)	\(\frac{ 5}{ 3}\)

Note:

Hadamard product $A \ast$

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

2

New Number: 2.62 | AESZ: 28 | Superseeker: 5 312 | Hash: 06dd455cafc5097e4f671d385984c1a2

Degree: 2

$\theta^4-x\left(65\theta^4+130\theta^3+105\theta^2+40\theta+6\right)+2^{2} x^{2}(4\theta+3)(\theta+1)^2(4\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 6, 126, 3948, 149310, ...
--> OEIS
Normalized instanton numbers (n₀=1): 5, 28, 312, 4808, 91048, ... ; Common denominator:...

Discriminant

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

Local exponents

$0$ $\frac{ 1}{ 64}$ $1$ $\infty$
$0$ $0$ $0$ $\frac{ 3}{ 4}$
$0$ $1$ $1$ $1$
$0$ $1$ $1$ $1$
$0$ $2$ $2$ $\frac{ 5}{ 4}$

\(0\)	\(\frac{ 1}{ 64}\)	\(1\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 3}{ 4}\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(2\)	\(2\)	\(\frac{ 5}{ 4}\)

Note:

A-incarnation: $X(1, 1, 1, 1, 1, 1) \subset Grass(3, 6)$

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-96\)	\(84\)	\(42\)	\(14\)

3

New Number: 11.13 | AESZ: | Superseeker: 70/13 15323/39 | Hash: 89df09ff1ec0d5dfcae0791579c9095e

Degree: 11

$13^{2} \theta^4-2 13 x\left(593\theta^4+850\theta^3+685\theta^2+260\theta+39\right)+2^{2} x^{2}\left(81227\theta^4+145178\theta^3+121774\theta^2+52312\theta+9477\right)-x^{3}\left(3180153\theta^4+8754414\theta^3+11733109\theta^2+7260552\theta+1687608\right)+2 x^{4}\left(9121117\theta^4+38823752\theta^3+61935546\theta^2+41745416\theta+10192764\right)-2^{2} x^{5}\left(14736265\theta^4+81359956\theta^3+152008790\theta^2+112521671\theta+29176827\right)+2^{2} 3^{2} x^{6}\left(1220244\theta^4+12211662\theta^3+31283769\theta^2+26817500\theta+7548762\right)+2^{2} 3^{2} x^{7}\left(4505067\theta^4+14797690\theta^3+6324743\theta^2-4986206\theta-2940402\right)-2^{3} 3^{3} x^{8}\left(855097\theta^4+3900198\theta^3+2679311\theta^2-619598\theta-662876\right)-2^{4} 3^{3} x^{9}\left(254021\theta^4+398518\theta^3+352691\theta^2+205022\theta+53940\right)+2^{5} 3^{3} 11 x^{10}\left(13283\theta^4+25990\theta^3+18039\theta^2+5062\theta+456\right)+2^{7} 3^{3} 11^{2} x^{11}(4\theta+3)(\theta+1)^2(4\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 6, 126, 4092, 160110, ...
--> OEIS
Normalized instanton numbers (n₀=1): 70/13, 420/13, 15323/39, 78225/13, 1564284/13, ... ; Common denominator:...

Discriminant

$(192z^2-69z+1)(2z^3+39z^2-5z+1)(13-112z-18z^2+132z^3)^2$

Local exponents

≈$-19.628663$ ≈$-0.912176$ $0$ $\frac{ 23}{ 128}-\frac{ 11}{ 384}\sqrt{ 33}$ ≈$0.064331-0.146063I$ ≈$0.064331+0.146063I$ ≈$0.115746$ $\frac{ 23}{ 128}+\frac{ 11}{ 384}\sqrt{ 33}$ ≈$0.932793$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 3}{ 4}$
$1$ $1$ $0$ $1$ $1$ $1$ $1$ $1$ $1$ $1$
$1$ $3$ $0$ $1$ $1$ $1$ $3$ $1$ $3$ $1$
$2$ $4$ $0$ $2$ $2$ $2$ $4$ $2$ $4$ $\frac{ 5}{ 4}$

≈\(-19.628663\)	≈\(-0.912176\)	\(0\)	\(\frac{ 23}{ 128}-\frac{ 11}{ 384}\sqrt{ 33}\)	≈\(0.064331-0.146063I\)	≈\(0.064331+0.146063I\)	≈\(0.115746\)	\(\frac{ 23}{ 128}+\frac{ 11}{ 384}\sqrt{ 33}\)	≈\(0.932793\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 3}{ 4}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(1\)	\(3\)	\(0\)	\(1\)	\(1\)	\(1\)	\(3\)	\(1\)	\(3\)	\(1\)
\(2\)	\(4\)	\(0\)	\(2\)	\(2\)	\(2\)	\(4\)	\(2\)	\(4\)	\(\frac{ 5}{ 4}\)

Note:

This is operator "11.13" from ...

\(-\frac{ 5}{ 16}+90\lambda\)	\(\frac{ 7}{ 32}+45\lambda\)	\(-\frac{ 7}{ 128}+\frac{ 15}{ 4}\lambda\)	\(.42666543e-1\)
\(-\frac{ 3}{ 8}\)	\(\frac{ 13}{ 16}\)	\(-\frac{ 1}{ 64}\)	\(-\frac{ 7}{ 128}-\frac{ 15}{ 4}\lambda\)
\(-\frac{ 9}{ 2}\)	\(\frac{ 3}{ 4}\)	\(\frac{ 13}{ 16}\)	\(\frac{ 7}{ 32}-45\lambda\)
\(-9\)	\(-\frac{ 9}{ 2}\)	\(-\frac{ 3}{ 8}\)	\(-\frac{ 5}{ 16}-90\lambda\)

\(16.+8.373926794I\)	\(-5-576\lambda\)	\(\frac{ 5}{ 4}+144\lambda\)	\(-.204864220-.332298682I\)
\(63\)	\(-20\)	\(\frac{ 21}{ 4}\)	\(-\frac{ 5}{ 4}-144\lambda\)
\(252\)	\(-84\)	\(22\)	\(-5-576\lambda\)
\(756+\frac{ 1}{ 500000000}I\)	\(-252\)	\(63\)	\(-14.-8.373926794I\)

Calabi-Yau differential operator database v.3

Summary

Discriminant

Local exponents

Note:

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

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

\(\frac{ 21}{ 16}-90\lambda\)	\(\frac{ 3}{ 8}\)	\(\frac{ 1}{ 2}\)	\(1\)
\(\frac{ 3}{ 8}\)	\(0\)	\(-1\)	\(0\)
\(\frac{ 9}{ 2}\)	\(-9\)	\(0\)	\(0\)
\(9\)	\(0\)	\(0\)	\(0\)

\(1+96\lambda\)	\(0\)	\(8\lambda\)	\(.5153046e-2\)
\(\frac{ 7}{ 2}\)	\(1\)	\(\frac{ 7}{ 24}\)	\(-8\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(42\)	\(0\)	\(\frac{ 7}{ 2}\)	\(1-96\lambda\)

\(-96\lambda\)	\(\frac{ 21}{ 2}\)	\(1\)	\(1\)
\(-\frac{ 7}{ 2}\)	\(-21\)	\(-1\)	\(0\)
\(0\)	\(42\)	\(0\)	\(0\)
\(-42\)	\(0\)	\(0\)	\(0\)