Calabi-Yau differential operator database v.3.0

1

New Number: 2.4 | AESZ: 62 | Superseeker: 372 71562236 | Hash: 07a3fd7577f878056e765831c6820f3d

Degree: 2

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 120, 138600, 228708480, 463140798120, ...
--> OEIS
Normalized instanton numbers (n₀=1): 372, 136182, 71562236, 63364481358, 65860679690400, ... ; Common denominator:...

Discriminant

$-(432z+1)(3456z-1)$

Local exponents

$-\frac{ 1}{ 432}$ $0$ $\frac{ 1}{ 3456}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 6}$
$1$ $0$ $1$ $\frac{ 5}{ 6}$
$1$ $0$ $1$ $\frac{ 7}{ 6}$
$2$ $0$ $2$ $\frac{ 11}{ 6}$

\(-\frac{ 1}{ 432}\)	\(0\)	\(\frac{ 1}{ 3456}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 6}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 5}{ 6}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 7}{ 6}\)
\(2\)	\(0\)	\(2\)	\(\frac{ 11}{ 6}\)

Note:

Hadamard product D*a

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-366\)	\(72\)	\(6\)	\(7\)

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: 5.107 | AESZ: 364 | Superseeker: 11/5 71/5 | Hash: c5b4bc60bc9d39ea420bd49fad182557

Degree: 5

$5^{2} \theta^4-5 x\left(553\theta^4+722\theta^3+611\theta^2+250\theta+40\right)+2^{6} x^{2}\left(1914\theta^4+4722\theta^3+5519\theta^2+3010\theta+610\right)-2^{12} x^{3}\left(685\theta^4+2400\theta^3+3466\theta^2+2220\theta+500\right)+2^{19} x^{4}(2\theta+1)(30\theta^3+105\theta^2+122\theta+46)-2^{25} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 120, 2240, 50680, ...
--> OEIS
Normalized instanton numbers (n₀=1): 11/5, -8/5, 71/5, 738, 26841/5, ... ; Common denominator:...

Discriminant

$-(32768z^3-2560z^2+85z-1)(-5+64z)^2$

Local exponents

$0$ ≈$0.023029$ ≈$0.027548-0.023797I$ ≈$0.027548+0.023797I$ $\frac{ 5}{ 64}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $1$ $1$ $1$
$0$ $1$ $1$ $1$ $3$ $1$
$0$ $2$ $2$ $2$ $4$ $\frac{ 3}{ 2}$

\(0\)	≈\(0.023029\)	≈\(0.027548-0.023797I\)	≈\(0.027548+0.023797I\)	\(\frac{ 5}{ 64}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)	\(3\)	\(1\)
\(0\)	\(2\)	\(2\)	\(2\)	\(4\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.107" from ...

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

4

New Number: 5.8 | AESZ: | Superseeker: 84 1522388/3 | Hash: f4b2a154823e983e64682b48f6254a15

Degree: 5

$\theta^4-2^{2} 3 x\left(192\theta^4+240\theta^3+191\theta^2+71\theta+10\right)+2^{7} 3^{2} x^{2}\left(1746\theta^4+3960\theta^3+4323\theta^2+2247\theta+395\right)-2^{12} 3^{4} x^{3}\left(2538\theta^4+7776\theta^3+9915\theta^2+5643\theta+1030\right)+2^{17} 3^{6} x^{4}\left(1782\theta^4+6480\theta^3+8793\theta^2+4905\theta+875\right)-2^{23} 3^{11} x^{5}(\theta+1)^2(3\theta+1)(3\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 120, 34920, 13157760, 5790070440, ...
--> OEIS
Normalized instanton numbers (n₀=1): 84, 9210, 1522388/3, 120348978, 19186016160, ... ; Common denominator:...

Discriminant

$-(-1+864z)(432z-1)^2(288z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 864}$ $\frac{ 1}{ 432}$ $\frac{ 1}{ 288}$ $\infty$
$0$ $0$ $0$ $0$ $\frac{ 1}{ 3}$
$0$ $1$ $\frac{ 1}{ 6}$ $1$ $1$
$0$ $1$ $\frac{ 5}{ 6}$ $3$ $1$
$0$ $2$ $1$ $4$ $\frac{ 5}{ 3}$

\(0\)	\(\frac{ 1}{ 864}\)	\(\frac{ 1}{ 432}\)	\(\frac{ 1}{ 288}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(0\)	\(1\)	\(\frac{ 1}{ 6}\)	\(1\)	\(1\)
\(0\)	\(1\)	\(\frac{ 5}{ 6}\)	\(3\)	\(1\)
\(0\)	\(2\)	\(1\)	\(4\)	\(\frac{ 5}{ 3}\)

Note:

This is operator "5.8" from ...

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

5

New Number: 8.55 | AESZ: | Superseeker: 1 68/9 | Hash: 39ed8ce7572bc79a333f77c892033bcf

Degree: 8

$\theta^4-x\left(33\theta^4+98\theta^3+105\theta^2+56\theta+12\right)+2^{3} x^{2}\left(34\theta^4+276\theta^3+609\theta^2+582\theta+216\right)+2^{4} 3 x^{3}\left(11\theta^4-170\theta^3-941\theta^2-1520\theta-846\right)-2^{7} 3^{2} x^{4}(2\theta^2+6\theta+5)(4\theta^2+12\theta-31)+2^{8} 3 x^{5}\left(11\theta^4+302\theta^3+1183\theta^2+1652\theta+726\right)+2^{11} x^{6}\left(34\theta^4+132\theta^3-39\theta^2-708\theta-747\right)-2^{12} x^{7}\left(33\theta^4+298\theta^3+1005\theta^2+1492\theta+816\right)+2^{16} x^{8}\left((\theta+3)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 120, 1216, 13080, ...
--> OEIS
Normalized instanton numbers (n₀=1): 1, -2, 68/9, -30, 150, ... ; Common denominator:...

Discriminant

$(z-1)(16z-1)(16z^2-16z+1)(4z-1)^2(4z+1)^2$

Local exponents

$-\frac{ 1}{ 4}$ $0$ $\frac{ 1}{ 16}$ $\frac{ 1}{ 2}-\frac{ 1}{ 4}\sqrt{ 3}$ $\frac{ 1}{ 4}$ $\frac{ 1}{ 2}+\frac{ 1}{ 4}\sqrt{ 3}$ $1$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $3$
$1$ $0$ $1$ $1$ $0$ $1$ $1$ $3$
$3$ $0$ $1$ $1$ $-1$ $1$ $1$ $3$
$4$ $0$ $2$ $2$ $1$ $2$ $2$ $3$

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

Note:

This is operator "8.55" from ...

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

6

New Number: 1.1 | AESZ: 1 | Superseeker: 575 63441275 | Hash: c86f1c284d8c5119801c6ba1343172bb

Degree: 1

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 120, 113400, 168168000, 305540235000, ...
--> OEIS
Normalized instanton numbers (n₀=1): 575, 121850, 63441275, 48493506000, 45861177777525, ... ; Common denominator:...

Discriminant

$1-3125z$

Local exponents

$0$ $\frac{ 1}{ 3125}$ $\infty$
$0$ $0$ $\frac{ 1}{ 5}$
$0$ $1$ $\frac{ 2}{ 5}$
$0$ $1$ $\frac{ 3}{ 5}$
$0$ $2$ $\frac{ 4}{ 5}$

\(0\)	\(\frac{ 1}{ 3125}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 5}\)
\(0\)	\(1\)	\(\frac{ 2}{ 5}\)
\(0\)	\(1\)	\(\frac{ 3}{ 5}\)
\(0\)	\(2\)	\(\frac{ 4}{ 5}\)

Note:

A-incarnation: $X(5) \subset P^4$
B-incarnation: mirror quintic.
P. Candelas, X. de la Ossa, D. Green, L. Parkes,{\em An exactly soluble superconformal theory from a mirror pair of Calabi-Yau manifolds}, Phys. Lett. B 258 (1991), no.1-2, 118-126.

\(4.+3.547288434I\)	\(-\frac{ 3}{ 2}-366\lambda\)	\(1.7500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000+2.069251586I\)	\(.298604603-366\lambda\)
\(7\)	\(-\frac{ 5}{ 2}\)	\(\frac{ 49}{ 12}\)	\(-1.7500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000-2.069251586I\)
\(6\)	\(-3\)	\(\frac{ 9}{ 2}\)	\(-\frac{ 3}{ 2}-366\lambda\)
\(12\)	\(-6\)	\(7\)	\(-2.-3.547288434I\)

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

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:

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:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Explicit solution

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

\(1+366\lambda\)	\(0\)	\(183\lambda\)	\(.524302301\)
\(3\)	\(1\)	\(\frac{ 3}{ 2}\)	\(-183\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(6\)	\(0\)	\(3\)	\(1-366\lambda\)

\(-366\lambda\)	\(4\)	\(1\)	\(1\)
\(-3\)	\(-3\)	\(-1\)	\(0\)
\(0\)	\(6\)	\(0\)	\(0\)
\(-6\)	\(0\)	\(0\)	\(0\)

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

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