Calabi-Yau differential operator database v.3.0

1

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

Degree: 2

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

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	,...

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

2

New Number: 5.26 | AESZ: 199 | Superseeker: -2 3820/9 | Hash: f7b5c9e3ad50b0885d03c98d07a051f1

Degree: 5

$\theta^4-x\left(15+88\theta+200\theta^2+224\theta^3+265\theta^4\right)+2 3 x^{2}\left(4325\theta^4+6386\theta^3+6011\theta^2+2718\theta+468\right)-2 3^{2} x^{3}\left(62015\theta^4+116478\theta^3+102361\theta^2+37422\theta+4824\right)+3^{6} 17 x^{4}\left(1465\theta^4+3092\theta^3+2686\theta^2+1140\theta+200\right)-3^{10} 17^{2} x^{5}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 15, 567, 28113, 1584279, ...
--> OEIS
Normalized instanton numbers (n₀=1): -2, 28, 3820/9, 3924, 21606, ... ; Common denominator:...

Discriminant

$-(z-1)(81z-1)^2(51z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 81}$ $\frac{ 1}{ 51}$ $1$ $\infty$
$0$ $0$ $0$ $0$ $1$
$0$ $\frac{ 1}{ 2}$ $1$ $1$ $1$
$0$ $\frac{ 1}{ 2}$ $3$ $1$ $1$
$0$ $1$ $4$ $2$ $1$

\(0\)	\(\frac{ 1}{ 81}\)	\(\frac{ 1}{ 51}\)	\(1\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(0\)	\(\frac{ 1}{ 2}\)	\(1\)	\(1\)	\(1\)
\(0\)	\(\frac{ 1}{ 2}\)	\(3\)	\(1\)	\(1\)
\(0\)	\(1\)	\(4\)	\(2\)	\(1\)

Note:

There is a second MUM-point at infinity, corresponding to
Operator AESZ 194/5.23.

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

3

New Number: 5.96 | AESZ: 339 | Superseeker: 12 28 | Hash: 41593acc689cf76c174442db98218947

Degree: 5

$\theta^4-2^{2} x\left(10\theta^4+50\theta^3+39\theta^2+14\theta+2\right)+2^{4} x^{2}\left(177\theta^4+1158\theta^3+2007\theta^2+1158\theta+230\right)+2^{8} x^{3}\left(539\theta^4+1344\theta^3-300\theta^2-1068\theta-340\right)+2^{10} 5 x^{4}(2\theta+1)(4\theta^3-642\theta^2-1002\theta-385)-2^{13} 3 5^{2} x^{5}(2\theta+1)(3\theta+2)(3\theta+4)(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 0, -6400, -249200, ...
--> OEIS
Normalized instanton numbers (n₀=1): 12, -339/2, 28, 27639/2, 634692, ... ; Common denominator:...

Discriminant

$-(55296z^3-5632z^2+80z-1)(1+20z)^2$

Local exponents

$-\frac{ 1}{ 20}$ $0$ ≈$0.007072-0.012497I$ ≈$0.007072+0.012497I$ ≈$0.087707$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$ $1$ $\frac{ 2}{ 3}$
$3$ $0$ $1$ $1$ $1$ $\frac{ 4}{ 3}$
$4$ $0$ $2$ $2$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 1}{ 20}\)	\(0\)	≈\(0.007072-0.012497I\)	≈\(0.007072+0.012497I\)	≈\(0.087707\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 2}{ 3}\)
\(3\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 4}{ 3}\)
\(4\)	\(0\)	\(2\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.96" from ...

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

Calabi-Yau differential operator database v.3

Summary

Discriminant

Local exponents

Note:

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

Discriminant

Local exponents

Note:

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

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

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

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