Calabi-Yau differential operator database v.3.0

1

New Number: 2.14 | AESZ: 48 | Superseeker: 24 5832 | Hash: 8081a3989d09a7d612953dac3341d90c

Degree: 2

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 24, 1800, 188160, 23423400, ...
--> OEIS
Normalized instanton numbers (n₀=1): 24, 291/2, 5832, 247116, 12634944, ... ; Common denominator:...

Discriminant

$(216z-1)(108z-1)$

Local exponents

$0$ $\frac{ 1}{ 216}$ $\frac{ 1}{ 108}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 3}$
$0$ $1$ $1$ $\frac{ 2}{ 3}$
$0$ $1$ $1$ $\frac{ 4}{ 3}$
$0$ $2$ $2$ $\frac{ 5}{ 3}$

\(0\)	\(\frac{ 1}{ 216}\)	\(\frac{ 1}{ 108}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 2}{ 3}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 4}{ 3}\)
\(0\)	\(2\)	\(2\)	\(\frac{ 5}{ 3}\)

Note:

B*d

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-162\)	\(84\)	\(24\)	\(11\)

2

New Number: 2.69 | AESZ: 205 | Superseeker: 1 5 | Hash: 4fb2e7002e630237d0458c3985cd6a18

Degree: 2

$\theta^4-x\left(59\theta^4+118\theta^3+105\theta^2+46\theta+8\right)+2^{5} 3 x^{2}(\theta+1)^2(3\theta+2)(3\theta+4)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 120, 2240, 46840, ...
--> OEIS
Normalized instanton numbers (n₀=1): 1, 7/4, 5, 24, 759/5, ... ; Common denominator:...

Discriminant

$(32z-1)(27z-1)$

Local exponents

$0$ $\frac{ 1}{ 32}$ $\frac{ 1}{ 27}$ $\infty$
$0$ $0$ $0$ $\frac{ 2}{ 3}$
$0$ $1$ $1$ $1$
$0$ $1$ $1$ $1$
$0$ $2$ $2$ $\frac{ 4}{ 3}$

\(0\)	\(\frac{ 1}{ 32}\)	\(\frac{ 1}{ 27}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 2}{ 3}\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(2\)	\(2\)	\(\frac{ 4}{ 3}\)

Note:

This is operator "2.69" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-128\)	\(160\)	\(160\)	\(40\)

3

New Number: 3.10 | AESZ: ~103 | Superseeker: 10 664 | Hash: 9239615e8ac132ca232c13367a39ae3b

Degree: 3

$\theta^4-2 x\left(86\theta^4+172\theta^3+143\theta^2+57\theta+9\right)+2^{2} 3^{2} x^{2}(\theta+1)^2(236\theta^2+472\theta+187)-2^{4} 3^{4} 5^{2} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 18, 630, 28980, 1593270, ...
--> OEIS
Normalized instanton numbers (n₀=1): 10, 24, 664, 9088, 234388, ... ; Common denominator:...

Discriminant

$-(100z-1)(-1+36z)^2$

Local exponents

$0$ $\frac{ 1}{ 100}$ $\frac{ 1}{ 36}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $\frac{ 1}{ 2}$ $1$
$0$ $1$ $\frac{ 1}{ 2}$ $2$
$0$ $2$ $1$ $\frac{ 5}{ 2}$

\(0\)	\(\frac{ 1}{ 100}\)	\(\frac{ 1}{ 36}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(\frac{ 1}{ 2}\)	\(1\)
\(0\)	\(1\)	\(\frac{ 1}{ 2}\)	\(2\)
\(0\)	\(2\)	\(1\)	\(\frac{ 5}{ 2}\)

Note:

Operator equivalent to AESZ 103 =$c \ast c$.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-48\)	\(72\)	\(36\)	\(12\)

4

New Number: 8.14 | AESZ: 176 | Superseeker: 24 15448/3 | Hash: e2a40a57f7e88dba6655d936b4abe327

Degree: 8

$\theta^4-2^{2} x(3\theta^2+3\theta+1)(17\theta^2+17\theta+6)+2^{5} x^{2}\left(325\theta^4+2164\theta^3+3053\theta^2+1778\theta+420\right)+2^{10} 3^{2} x^{3}\left(51\theta^4-306\theta^3-934\theta^2-717\theta-204\right)-2^{14} 3^{2} x^{4}\left(397\theta^4+794\theta^3-1454\theta^2-1851\theta-666\right)+2^{18} 3^{4} x^{5}\left(51\theta^4+510\theta^3+290\theta^2-29\theta-64\right)+2^{21} 3^{4} x^{6}\left(325\theta^4-864\theta^3-1489\theta^2-864\theta-144\right)-2^{26} 3^{6} x^{7}(3\theta^2+3\theta+1)(17\theta^2+17\theta+6)+2^{32} 3^{8} x^{8}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 24, 840, 34944, 1618344, ...
--> OEIS
Normalized instanton numbers (n₀=1): 24, -509/2, 15448/3, -128530, 3746624, ... ; Common denominator:...

Discriminant

$(72z-1)(36z-1)(64z-1)(32z-1)(48z-1)^2(48z+1)^2$

Local exponents

$-\frac{ 1}{ 48}$ $0$ $\frac{ 1}{ 72}$ $\frac{ 1}{ 64}$ $\frac{ 1}{ 48}$ $\frac{ 1}{ 36}$ $\frac{ 1}{ 32}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $0$ $1$ $1$ $1$ $1$ $1$ $1$
$3$ $0$ $1$ $1$ $3$ $1$ $1$ $1$
$4$ $0$ $2$ $2$ $4$ $2$ $2$ $1$

\(-\frac{ 1}{ 48}\)	\(0\)	\(\frac{ 1}{ 72}\)	\(\frac{ 1}{ 64}\)	\(\frac{ 1}{ 48}\)	\(\frac{ 1}{ 36}\)	\(\frac{ 1}{ 32}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(3\)	\(0\)	\(1\)	\(1\)	\(3\)	\(1\)	\(1\)	\(1\)
\(4\)	\(0\)	\(2\)	\(2\)	\(4\)	\(2\)	\(2\)	\(1\)

Note:

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

\(\frac{ 13}{ 4}+324\lambda\)	\(-\frac{ 9}{ 16}-81\lambda\)	\(\frac{ 21}{ 64}+\frac{ 189}{ 4}\lambda\)	\(-.54109387e-1-\frac{ 243}{ 8}\lambda\)
\(7+\frac{ 1}{ 500000000}I\)	\(-\frac{ 3}{ 4}+\frac{ 1}{ 1000000000}I\)	\(\frac{ 49}{ 48}-\frac{ 1}{ 1000000000}I\)	\(-\frac{ 21}{ 64}-\frac{ 189}{ 4}\lambda\)
\(12+\frac{ 1}{ 250000000}I\)	\(-3+\frac{ 1}{ 500000000}I\)	\(\frac{ 11}{ 4}-\frac{ 1}{ 500000000}I\)	\(-\frac{ 9}{ 16}-81\lambda\)
\(48.+.19e-7I\)	\(-11.999999983+.80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e-8I\)	\(7-\frac{ 1}{ 125000000}I\)	\(-\frac{ 5}{ 4}-324\lambda\)

\(3.333327178+1.235680918I\)	\(-.293785798-.155290034I\)	\(.97173439e-1+.52174089e-1I\)	\(-.12126753e-1-.18089273e-1I\)
\(13.327492354-.18865422e-1I\)	\(-.677351606+.3680160e-2I\)	\(.556724039+.2392839e-2I\)	\(-.96900942e-1-.51907518e-1I\)
\(39.969803488-.74697659e-1I\)	\(-5.030457990+.13317419e-1I\)	\(2.669647034+\frac{ 1}{ 243}IPi^{ \frac{ 1}{ 6}}Zeta(5)^7\)	\(-.290681436-.15554156171284634760705289672544080604534005037783375314861460957178841309823677581863979848866498740554156171284634760705289672544080604534005037783375314861460957178841309823677581863979848866498741I\)
\(319.859816505-.452770130I\)	\(-40.256438534+.88323845e-1I\)	\(13.361376945+.57428145e-1I\)	\(-1.325622606-1.245780435I\)

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:

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

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

\(1+162\lambda\)	\(0\)	\(\frac{ 189}{ 8}\lambda\)	\(.25679681e-1\)
\(\frac{ 7}{ 2}\)	\(1\)	\(\frac{ 49}{ 96}\)	\(-\frac{ 189}{ 8}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(24\)	\(0\)	\(\frac{ 7}{ 2}\)	\(1-162\lambda\)

\(-162\lambda\)	\(\frac{ 15}{ 2}\)	\(1\)	\(1\)
\(-\frac{ 7}{ 2}\)	\(-12\)	\(-1\)	\(0\)
\(0\)	\(24\)	\(0\)	\(0\)
\(-24\)	\(0\)	\(0\)	\(0\)

\(1.+.620290874I\)	\(0\)	\(\frac{ 16}{ 3}\lambda\)	\(.2404755e-2\)
\(\frac{ 20}{ 3}\)	\(1\)	\(\frac{ 5}{ 18}\)	\(-\frac{ 16}{ 3}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(160\)	\(0\)	\(\frac{ 20}{ 3}\)	\(1.-.620290874I\)

\(-\frac{ 310145437}{ 500000000}I\)	\(\frac{ 100}{ 3}\)	\(1\)	\(1\)
\(-\frac{ 20}{ 3}\)	\(-80\)	\(-1\)	\(0\)
\(0\)	\(160\)	\(0\)	\(0\)
\(-160\)	\(0\)	\(0\)	\(0\)

\(-48\lambda\)	\(9\)	\(1\)	\(1\)
\(-3\)	\(-18\)	\(-1\)	\(0\)
\(0\)	\(36\)	\(0\)	\(0\)
\(-36\)	\(0\)	\(0\)	\(0\)