Calabi-Yau differential operator database v.3.0

1

New Number: 2.24 | AESZ: 137 | Superseeker: 20 1684/3 | Hash: 198d6c822d6c46225ac2553d60df6539

Degree: 2

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

Coefficients of the holomorphic solution: 1, 24, 1512, 124800, 11730600, ...
--> OEIS
Normalized instanton numbers (n₀=1): 20, 2, 1684/3, 7602, 173472, ... ; Common denominator:...

Discriminant

$(144z-1)(128z-1)$

Local exponents

$0$ $\frac{ 1}{ 144}$ $\frac{ 1}{ 128}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $\frac{ 3}{ 2}$
$0$ $2$ $2$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 1}{ 144}\)	\(\frac{ 1}{ 128}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 3}{ 2}\)
\(0\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

Hadamard product $A \ast g$.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-16\)	\(96\)	\(48\)	\(16\)

2

New Number: 2.52 | AESZ: 16 | Superseeker: 4 644/3 | Hash: 05af0662662bfbec63e3186c4f363313

Degree: 2

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 168, 5120, 190120, ...
--> OEIS
Normalized instanton numbers (n₀=1): 4, 20, 644/3, 3072, 52512, ... ; Common denominator:...

Discriminant

$(64z-1)(16z-1)$

Local exponents

$0$ $\frac{ 1}{ 64}$ $\frac{ 1}{ 16}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $1$
$0$ $1$ $1$ $1$
$0$ $2$ $2$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 1}{ 64}\)	\(\frac{ 1}{ 16}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

Hadamard product $I \ast \alpha$
A-Incarnation: diagonal subfamily of 1,1,1,1-intersection in $P^1 \times P^1 \times P^1 \times \P^1$
B-Incarnations:
Fibre products: 62211- x 632--1, S62211

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-128\)	\(96\)	\(48\)	\(16\)

3

New Number: 5.9 | AESZ: 56 | Superseeker: -16 -3280 | Hash: 58a7f24bf18cb98b526885069667f9f0

Degree: 5

$\theta^4-2^{4} x\left(22\theta^4+8\theta^3+9\theta^2+5\theta+1\right)+2^{9} x^{2}\left(94\theta^4+88\theta^3+97\theta^2+45\theta+8\right)-2^{14} x^{3}\left(194\theta^4+336\theta^3+371\theta^2+195\theta+41\right)+2^{19} 3 x^{4}\left(64\theta^4+176\theta^3+217\theta^2+129\theta+30\right)-2^{27} 3^{2} x^{5}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 16, 464, 17152, 725776, ...
--> OEIS
Normalized instanton numbers (n₀=1): -16, -178, -3280, -76197, -2046896, ... ; Common denominator:...

Discriminant

$-(-1+32z)(96z-1)^2(64z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 96}$ $\frac{ 1}{ 64}$ $\frac{ 1}{ 32}$ $\infty$
$0$ $0$ $0$ $0$ $1$
$0$ $1$ $\frac{ 1}{ 2}$ $1$ $1$
$0$ $3$ $\frac{ 1}{ 2}$ $1$ $1$
$0$ $4$ $1$ $2$ $1$

\(0\)	\(\frac{ 1}{ 96}\)	\(\frac{ 1}{ 64}\)	\(\frac{ 1}{ 32}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(0\)	\(1\)	\(\frac{ 1}{ 2}\)	\(1\)	\(1\)
\(0\)	\(3\)	\(\frac{ 1}{ 2}\)	\(1\)	\(1\)
\(0\)	\(4\)	\(1\)	\(2\)	\(1\)

Note:

There is a second MUM-point hiding at infinity, corresponding to Operator...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(544\)	\(-64\)	\(128\)	\(16\)

4

New Number: 7.1 | AESZ: | Superseeker: 10/7 508/7 | Hash: 08ab3cb496250adfa30bc3e24ac63c4f

Degree: 7

$7^{2} \theta^4-2 7 x\theta(46\theta^3+52\theta^2+33\theta+7)-2^{2} x^{2}\left(7332\theta^4+28848\theta^3+42633\theta^2+26670\theta+6272\right)-2^{4} x^{3}\left(2860\theta^4+44760\theta^3+120483\theta^2+111279\theta+35098\right)+2^{9} x^{4}\left(2230\theta^4+5920\theta^3-741\theta^2-6509\theta-3049\right)+2^{14} x^{5}\left(174\theta^4+1320\theta^3+1971\theta^2+1095\theta+190\right)-2^{19} x^{6}\left(22\theta^4+24\theta^3-9\theta^2-21\theta-7\right)-2^{25} x^{7}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 32, 288, 7776, ...
--> OEIS
Normalized instanton numbers (n₀=1): 10/7, 100/7, 508/7, 808, 59910/7, ... ; Common denominator:...

Discriminant

$-(16z+1)(32z-1)(32z-7)^2(4z+1)^3$

Local exponents

$-\frac{ 1}{ 4}$ $-\frac{ 1}{ 16}$ $0$ $\frac{ 1}{ 32}$ $\frac{ 7}{ 32}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $1$
$\frac{ 1}{ 2}$ $1$ $0$ $1$ $1$ $1$
$\frac{ 3}{ 2}$ $1$ $0$ $1$ $3$ $1$
$2$ $2$ $0$ $2$ $4$ $1$

\(-\frac{ 1}{ 4}\)	\(-\frac{ 1}{ 16}\)	\(0\)	\(\frac{ 1}{ 32}\)	\(\frac{ 7}{ 32}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(\frac{ 1}{ 2}\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(\frac{ 3}{ 2}\)	\(1\)	\(0\)	\(1\)	\(3\)	\(1\)
\(2\)	\(2\)	\(0\)	\(2\)	\(4\)	\(1\)

Note:

This is operator "7.1" from ...

\(.924152876+.943952044I\)	\(.188111204+1.275393351I\)	\(.593885779-.164736576I\)	\(-.98788356e-1-.111719004I\)
\(.205812544+3.214663066I\)	\(2.256285526+4.201440782I\)	\(1.918711704-.844672672I\)	\(-.387126151-.327818207I\)
\(-.81202836e-1+5.639775328I\)	\(1.617313859+7.506248400I\)	\(4.465075410-1.211574339I\)	\(-.630856884-.62528948587308939323761000463177396943029180176007410838351088466882816118573413617415470125057897174617878647522000926354793886058360352014821676702176933765632237146827234830940250115794349235757295I\)
\(2.469750531+38.575956789I\)	\(15.075426310+50.417289384I\)	\(23.024540442-10.136072063I\)	\(-3.645513812-3.933818487I\)

\(5+512\lambda\)	\(-1.-.620290874I\)	\(\frac{ 1}{ 3}+\frac{ 128}{ 3}\lambda\)	\(-\frac{ 1}{ 7}7^{ \frac{ 5}{ 8})exp(-\frac{ 9}{ 4}}Zeta(5)^(\frac{ 1}{ 4})-\frac{ 64}{ 3}\lambda\)
\(16\)	\(-3\)	\(\frac{ 4}{ 3}\)	\(-\frac{ 1}{ 3}-\frac{ 128}{ 3}\lambda\)
\(48\)	\(-12\)	\(5\)	\(-1.-.620290874I\)
\(192\)	\(-48\)	\(16\)	\(-3-512\lambda\)

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

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

\(1+16\lambda\)	\(0\)	\(\frac{ 4}{ 3}\lambda\)	\(.125248e-3\)
\(4\)	\(1\)	\(\frac{ 1}{ 3}\)	\(-\frac{ 4}{ 3}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(48\)	\(0\)	\(4\)	\(1-16\lambda\)

\(-16\lambda\)	\(12\)	\(1\)	\(1\)
\(-4\)	\(-24\)	\(-1\)	\(0\)
\(0\)	\(48\)	\(0\)	\(0\)
\(-48\)	\(0\)	\(0\)	\(0\)

\(1.+.620290874I\)	\(0\)	\(\frac{ 32}{ 3}\lambda\)	\(.8015849e-2\)
\(4\)	\(1\)	\(\frac{ 1}{ 3}\)	\(-\frac{ 32}{ 3}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(48\)	\(0\)	\(4\)	\(1.-.620290874I\)

\(-\frac{ 310145437}{ 500000000}I\)	\(12\)	\(1\)	\(1\)
\(-4\)	\(-24\)	\(-1\)	\(0\)
\(0\)	\(48\)	\(0\)	\(0\)
\(-48\)	\(0\)	\(0\)	\(0\)