Calabi-Yau differential operator database v.3.0

1

New Number: 2.1 | AESZ: 45 | Superseeker: 12 3204 | Hash: cdf289f6febf84eb577a238542a57457

Degree: 2

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

Coefficients of the holomorphic solution: 1, 8, 360, 22400, 1695400, ...
--> OEIS
Normalized instanton numbers (n₀=1): 12, 163, 3204, 107582, 4203360, ... ; Common denominator:...

Discriminant

$-(16z+1)(128z-1)$

Local exponents

$-\frac{ 1}{ 16}$ $0$ $\frac{ 1}{ 128}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $\frac{ 3}{ 2}$
$2$ $0$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 1}{ 16}\)	\(0\)	\(\frac{ 1}{ 128}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 3}{ 2}\)
\(2\)	\(0\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

Hadamard product $A \ast a$, where $A$ is (:case 2.1.1)

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-120\)	\(72\)	\(24\)	\(10\)

2

New Number: 2.27 | AESZ: 140 | Superseeker: 108 -4945756 | Hash: 74692097f8183c067c2c4b1a5c93387b

Degree: 2

$\theta^4-2^{2} 3 x(6\theta+1)(6\theta+5)(17\theta^2+17\theta+6)+2^{7} 3^{4} x^{2}(6\theta+1)(6\theta+5)(6\theta+7)(6\theta+11)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 360, 582120, 1274232960, 3204505984680, ...
--> OEIS
Normalized instanton numbers (n₀=1): 108, 54135, -4945756, 7925523138, -2434666062240, ... ; Common denominator:...

Discriminant

$(3888z-1)(3456z-1)$

Local exponents

$0$ $\frac{ 1}{ 3888}$ $\frac{ 1}{ 3456}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 6}$
$0$ $1$ $1$ $\frac{ 5}{ 6}$
$0$ $1$ $1$ $\frac{ 7}{ 6}$
$0$ $2$ $2$ $\frac{ 11}{ 6}$

\(0\)	\(\frac{ 1}{ 3888}\)	\(\frac{ 1}{ 3456}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 6}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 5}{ 6}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 7}{ 6}\)
\(0\)	\(2\)	\(2\)	\(\frac{ 11}{ 6}\)

Note:

Hadamard product $D \ast g$

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-676\)	\(132\)	\(12\)	\(13\)

3

New Number: 5.88 | AESZ: 324 | Superseeker: 148/11 44108/11 | Hash: 7f84d776cf00ff399b20865542185f87

Degree: 5

$11^{2} \theta^4-2^{2} 11 x\left(432\theta^4+624\theta^3+477\theta^2+165\theta+22\right)+2^{5} x^{2}\left(12944\theta^4+4736\theta^3-15491\theta^2-12914\theta-2860\right)-2^{4} 5 x^{3}\left(10688\theta^4-114048\theta^3-159132\theta^2-83028\theta-15455\right)-2^{11} 5^{2} x^{4}(2\theta+1)(4\theta+3)(76\theta^2+189\theta+125)+2^{14} 5^{3} x^{5}(2\theta+1)(4\theta+3)(4\theta+5)(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 360, 23120, 1796200, ...
--> OEIS
Normalized instanton numbers (n₀=1): 148/11, 2044/11, 44108/11, 1459636/11, 60212712/11, ... ; Common denominator:...

Discriminant

$(5120z^3-512z^2-128z+1)(-11+160z)^2$

Local exponents

≈$-0.120643$ $0$ ≈$0.007599$ $\frac{ 11}{ 160}$ ≈$0.213044$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$ $1$ $\frac{ 3}{ 4}$
$1$ $0$ $1$ $3$ $1$ $\frac{ 5}{ 4}$
$2$ $0$ $2$ $4$ $2$ $\frac{ 3}{ 2}$

≈\(-0.120643\)	\(0\)	≈\(0.007599\)	\(\frac{ 11}{ 160}\)	≈\(0.213044\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 3}{ 4}\)
\(1\)	\(0\)	\(1\)	\(3\)	\(1\)	\(\frac{ 5}{ 4}\)
\(2\)	\(0\)	\(2\)	\(4\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.88" from ...

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

4

New Number: 1.8 | AESZ: 8 | Superseeker: 2628 3966805740 | Hash: 1a7187fdf63fe8761c969fdab1af1c36

Degree: 1

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 360, 1247400, 6861254400, 46381007673000, ...
--> OEIS
Normalized instanton numbers (n₀=1): 2628, 2009484, 3966805740, 11533584001896, 41531678111043360, ... ; Common denominator:...

Discriminant

$1-11664z$

Local exponents

$0$ $\frac{ 1}{ 11664}$ $\infty$
$0$ $0$ $\frac{ 1}{ 6}$
$0$ $1$ $\frac{ 1}{ 3}$
$0$ $1$ $\frac{ 2}{ 3}$
$0$ $2$ $\frac{ 5}{ 6}$

\(0\)	\(\frac{ 1}{ 11664}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 6}\)
\(0\)	\(1\)	\(\frac{ 1}{ 3}\)
\(0\)	\(1\)	\(\frac{ 2}{ 3}\)
\(0\)	\(2\)	\(\frac{ 5}{ 6}\)

Note:

A-incarnation of $X(6) \subset P^4(1,1,1,1,2)$.

\(4+240\lambda\)	\(-\frac{ 3}{ 2}-120\lambda\)	\(\frac{ 5}{ 8}+50\lambda\)	\(-.159319280-30\lambda\)
\(10\)	\(-4\)	\(\frac{ 25}{ 12}\)	\(-\frac{ 5}{ 8}-50\lambda\)
\(24\)	\(-12\)	\(6\)	\(-\frac{ 3}{ 2}-120\lambda\)
\(48\)	\(-24\)	\(10\)	\(-2-240\lambda\)

\(-5.088262696-2.403669772I\)	\(-9.231233148+8.267057761I\)	\(6.171205313+7.544982974I\)	\(3.227450678-3.068539845I\)
\(-6.270037138+8.225054977I\)	\(9.605336680+17.587574161I\)	\(14.123297626-6.142790642I\)	\(-3.279565479-6.225556868I\)
\(-2.195545276+2.457029703I\)	\(2.297135143+5.799775915I\)	\(5.638341607-1.600871223I\)	\(-.887292619-2.058764958I\)
\(-13.680081028+17.945574495I\)	\(18.775280028+38.372889078I\)	\(30.814467548-13.402452309I\)	\(-6.155415591-13.583033166I\)

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:

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

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

\(1+120\lambda\)	\(0\)	\(15\lambda\)	\(.14090360e-1\)
\(3\)	\(1\)	\(\frac{ 3}{ 8}\)	\(-15\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(24\)	\(0\)	\(3\)	\(1-120\lambda\)

\(-120\lambda\)	\(7\)	\(1\)	\(1\)
\(-3\)	\(-12\)	\(-1\)	\(0\)
\(0\)	\(24\)	\(0\)	\(0\)
\(-24\)	\(0\)	\(0\)	\(0\)

\(1.+3.275911176I\)	\(0\)	\(1.501459289I\)	\(.894299503\)
\(\frac{ 11}{ 2}\)	\(1\)	\(\frac{ 121}{ 48}\)	\(-1.501459289I\)
\(0\)	\(0\)	\(1\)	\(0\)
\(12\)	\(0\)	\(\frac{ 11}{ 2}\)	\(1.-3.275911176I\)

\(-\frac{ 409488897}{ 125000000}I\)	\(\frac{ 15}{ 2}\)	\(1\)	\(1\)
\(-\frac{ 11}{ 2}\)	\(-6\)	\(-1\)	\(0\)
\(0\)	\(12\)	\(0\)	\(0\)
\(-12\)	\(0\)	\(0\)	\(0\)