Calabi-Yau differential operator database v.3.0

1

New Number: 2.61 | AESZ: 26 | Superseeker: 10 1724 | Hash: f3fc09474973b19b8bdb783e3322eb65

Degree: 2

$\theta^4-2 x(2\theta+1)^2(13\theta^2+13\theta+4)-2^{2} 3 x^{2}(2\theta+1)(3\theta+2)(3\theta+4)(2\theta+3)$

Coefficients of the holomorphic solution: 1, 8, 288, 15200, 968800, ...
--> OEIS
Normalized instanton numbers (n₀=1): 10, 191/2, 1724, 45680, 1478214, ... ; Common denominator:...

Discriminant

$-(4z+1)(108z-1)$

Local exponents

$-\frac{ 1}{ 4}$ $0$ $\frac{ 1}{ 108}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $\frac{ 2}{ 3}$
$1$ $0$ $1$ $\frac{ 4}{ 3}$
$2$ $0$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 1}{ 4}\)	\(0\)	\(\frac{ 1}{ 108}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 2}{ 3}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 4}{ 3}\)
\(2\)	\(0\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

A-incarnation: $X(1,1,1,1,2) \subset Grass(2,6)$

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-116\)	\(76\)	\(28\)	\(11\)

2

New Number: 5.23 | AESZ: 194 | Superseeker: 126/17 11700/17 | Hash: 6bf19665aa6705f30ef88df42bc4eac4

Degree: 5

$17^{2} \theta^4-17 x\left(1465\theta^4+2768\theta^3+2200\theta^2+816\theta+119\right)+2 x^{2}\left(62015\theta^4+131582\theta^3+125017\theta^2+65926\theta+15300\right)-2 3^{3} x^{3}\left(4325\theta^4+10914\theta^3+12803\theta^2+7446\theta+1700\right)+3^{6} x^{4}\left(265\theta^4+836\theta^3+1118\theta^2+700\theta+168\right)-3^{10} x^{5}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 7, 183, 7225, 345079, ...
--> OEIS
Normalized instanton numbers (n₀=1): 126/17, 848/17, 11700/17, 229808/17, 5539258/17, ... ; Common denominator:...

Discriminant

$-(-1+81z)(27z-17)^2(z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 81}$ $\frac{ 17}{ 27}$ $1$ $\infty$
$0$ $0$ $0$ $0$ $1$
$0$ $1$ $1$ $\frac{ 1}{ 2}$ $1$
$0$ $1$ $3$ $\frac{ 1}{ 2}$ $1$
$0$ $2$ $4$ $1$ $1$

\(0\)	\(\frac{ 1}{ 81}\)	\(\frac{ 17}{ 27}\)	\(1\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(0\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)	\(1\)
\(0\)	\(1\)	\(3\)	\(\frac{ 1}{ 2}\)	\(1\)
\(0\)	\(2\)	\(4\)	\(1\)	\(1\)

Note:

There is a second MUM-point at infinity, corresponding to Operator AESZ 199/5.26

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-88\)	\(76\)	\(34\)	\(12\)

3

New Number: 8.20 | AESZ: 213 | Superseeker: 118/17 672 | Hash: d430b37f4ca641af0b82cbef83547c51

Degree: 8

$17^{2} \theta^4-2 17 x\left(647\theta^4+1240\theta^3+977\theta^2+357\theta+51\right)-2^{2} x^{2}\left(14437\theta^4+89752\theta^3+147734\theta^2+92123\theta+20400\right)+2^{2} 3 x^{3}\left(21538\theta^4+25680\theta^3-41979\theta^2-56151\theta-17442\right)+2^{3} x^{4}\left(51920\theta^4+166384\theta^3-83149\theta^2-217017\theta-79362\right)-2^{4} 3 x^{5}\left(9360\theta^4-26784\theta^3-43813\theta^2-21965\theta-3496\right)-2^{5} 3 x^{6}\left(10160\theta^4-96\theta^3-10535\theta^2-5385\theta-438\right)-2^{8} 3^{2} x^{7}\left(288\theta^4+864\theta^3+1082\theta^2+641\theta+147\right)-2^{11} 3^{2} x^{8}(4\theta+3)(\theta+1)^2(4\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 6, 162, 6252, 290610, ...
--> OEIS
Normalized instanton numbers (n₀=1): 118/17, 873/17, 672, 447987/34, 5358846/17, ... ; Common denominator:...

Discriminant

$-(4z+1)(32z^3+40z^2+78z-1)(-17+18z+48z^2)^2$

Local exponents

$-\frac{ 3}{ 16}-\frac{ 1}{ 48}\sqrt{ 897}$ ≈$-0.631368-1.433512I$ ≈$-0.631368+1.433512I$ $-\frac{ 1}{ 4}$ $0$ ≈$0.012736$ $-\frac{ 3}{ 16}+\frac{ 1}{ 48}\sqrt{ 897}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 3}{ 4}$
$1$ $1$ $1$ $1$ $0$ $1$ $1$ $1$
$3$ $1$ $1$ $1$ $0$ $1$ $3$ $1$
$4$ $2$ $2$ $2$ $0$ $2$ $4$ $\frac{ 5}{ 4}$

\(-\frac{ 3}{ 16}-\frac{ 1}{ 48}\sqrt{ 897}\)	≈\(-0.631368-1.433512I\)	≈\(-0.631368+1.433512I\)	\(-\frac{ 1}{ 4}\)	\(0\)	≈\(0.012736\)	\(-\frac{ 3}{ 16}+\frac{ 1}{ 48}\sqrt{ 897}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 3}{ 4}\)
\(1\)	\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(3\)	\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(3\)	\(1\)
\(4\)	\(2\)	\(2\)	\(2\)	\(0\)	\(2\)	\(4\)	\(\frac{ 5}{ 4}\)

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-90\)	\(76\)	\(34\)	\(12\)

4

New Number: 8.7 | AESZ: 106 | Superseeker: 12 356 | Hash: fe1c90929d18b81637eaaa93366409ed

Degree: 8

$\theta^4-2^{2} x(3\theta^2+3\theta+1)(11\theta^2+11\theta+3)+2^{4} x^{2}\left(241\theta^4+940\theta^3+1303\theta^2+726\theta+145\right)-2^{7} x^{3}\left(33\theta^4-198\theta^3-607\theta^2-456\theta-117\right)+2^{10} x^{4}\left(239\theta^4+478\theta^3-322\theta^2-561\theta-169\right)+2^{12} x^{5}\left(33\theta^4+330\theta^3+185\theta^2-32\theta-37\right)+2^{14} x^{6}\left(241\theta^4+24\theta^3-71\theta^2+24\theta+23\right)+2^{17} x^{7}(3\theta^2+3\theta+1)(11\theta^2+11\theta+3)+2^{20} x^{8}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 380, 16464, 845676, ...
--> OEIS
Normalized instanton numbers (n₀=1): 12, 20, 356, 34561/4, 161840, ... ; Common denominator:...

Discriminant

$(64z^2+88z-1)(16z^2+44z-1)(1+32z^2)^2$

Local exponents

$-\frac{ 11}{ 8}-\frac{ 5}{ 8}\sqrt{ 5}$ $-\frac{ 11}{ 16}-\frac{ 5}{ 16}\sqrt{ 5}$ $0-\frac{ 1}{ 8}\sqrt{ 2}I$ $0$ $0+\frac{ 1}{ 8}\sqrt{ 2}I$ $-\frac{ 11}{ 16}+\frac{ 5}{ 16}\sqrt{ 5}$ $-\frac{ 11}{ 8}+\frac{ 5}{ 8}\sqrt{ 5}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $0$ $1$ $1$ $1$ $1$
$1$ $1$ $3$ $0$ $3$ $1$ $1$ $1$
$2$ $2$ $4$ $0$ $4$ $2$ $2$ $1$

\(-\frac{ 11}{ 8}-\frac{ 5}{ 8}\sqrt{ 5}\)	\(-\frac{ 11}{ 16}-\frac{ 5}{ 16}\sqrt{ 5}\)	\(0-\frac{ 1}{ 8}\sqrt{ 2}I\)	\(0\)	\(0+\frac{ 1}{ 8}\sqrt{ 2}I\)	\(-\frac{ 11}{ 16}+\frac{ 5}{ 16}\sqrt{ 5}\)	\(-\frac{ 11}{ 8}+\frac{ 5}{ 8}\sqrt{ 5}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(3\)	\(0\)	\(3\)	\(1\)	\(1\)	\(1\)
\(2\)	\(2\)	\(4\)	\(0\)	\(4\)	\(2\)	\(2\)	\(1\)

Note:

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

\(\frac{ 20}{ 3}+464\lambda\)	\(-\frac{ 17}{ 6}-232\lambda\)	\(1.0456349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349+.414911827I\)	\(-.241563519-.227532292I\)
\(\frac{ 62}{ 3}\)	\(-\frac{ 28}{ 3}\)	\(\frac{ 961}{ 252}\)	\(-1.0456349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349-.414911827I\)
\(56\)	\(-28\)	\(\frac{ 34}{ 3}\)	\(-\frac{ 17}{ 6}-232\lambda\)
\(112\)	\(-56\)	\(\frac{ 62}{ 3}\)	\(-\frac{ 14}{ 3}-464\lambda\)

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:

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

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

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

\(1+116\lambda\)	\(0\)	\(.63575199e-1I\)	\(\frac{ 79}{ 7000}\)
\(\frac{ 19}{ 6}\)	\(1\)	\(\frac{ 361}{ 1008}\)	\(-.63575199e-1I\)
\(0\)	\(0\)	\(1\)	\(0\)
\(28\)	\(0\)	\(\frac{ 19}{ 6}\)	\(1-116\lambda\)

\(-116\lambda\)	\(\frac{ 47}{ 6}\)	\(1\)	\(1\)
\(-\frac{ 19}{ 6}\)	\(-14\)	\(-1\)	\(0\)
\(0\)	\(28\)	\(0\)	\(0\)
\(-28\)	\(0\)	\(0\)	\(0\)