Calabi-Yau differential operator database v.3.0

1

New Number: 4.1 | AESZ: ~39 | Superseeker: -32 -8736 | Hash: 462066f711fc3742db1ea9befa2fe01b

Degree: 4

$\theta^4-2^{2} x\left(160\theta^4+320\theta^3+386\theta^2+226\theta+51\right)+2^{4} 3 x^{2}\left(2816\theta^4+11264\theta^3+19360\theta^2+16192\theta+5491\right)-2^{15} x^{3}(80\theta^2+240\theta+243)(2\theta+3)^2+2^{26} x^{4}(\theta+2)^2(2\theta+3)(2\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 204, 41820, 9022160, 2025179100, ...
--> OEIS
Normalized instanton numbers (n₀=1): -32, -284, -8736, -283900, -10041888, ... ; Common denominator:...

Discriminant

$(256z-1)^2(64z-1)^2$

Local exponents

$0$ $\infty$ $\frac{ 1}{ 256}$ $\frac{ 1}{ 64}$
$0$ $\frac{ 3}{ 2}$ $0$ $0$
$0$ $2$ $-\frac{ 1}{ 2}$ $-\frac{ 1}{ 2}$
$0$ $2$ $1$ $1$
$0$ $\frac{ 5}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$

\(0\)	\(\infty\)	\(\frac{ 1}{ 256}\)	\(\frac{ 1}{ 64}\)
\(0\)	\(\frac{ 3}{ 2}\)	\(0\)	\(0\)
\(0\)	\(2\)	\(-\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 2}\)
\(0\)	\(2\)	\(1\)	\(1\)
\(0\)	\(\frac{ 5}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)

Note:

YY-Operator equivalent to AESZ 39=$A \ast \alpha$.

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

2

New Number: 4.2 | AESZ: ~44 | Superseeker: -76 -92996 | Hash: 79f5f70bb79e740c1cd7e835ff99a64c

Degree: 4

$\theta^4-2^{2} x\left(272\theta^4+544\theta^3+649\theta^2+377\theta+84\right)+2^{6} 3 x^{2}\left(1544\theta^4+6176\theta^3+9307\theta^2+6262\theta+1588\right)-2^{8} x^{3}(272\theta^2+816\theta+819)(2\theta+3)^2+2^{14} x^{4}(\theta+2)^2(2\theta+3)(2\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 336, 142728, 65762368, 31568339880, ...
--> OEIS
Normalized instanton numbers (n₀=1): -76, -2002, -92996, -5555506, -384650592, ... ; Common denominator:...

Discriminant

$(1-544z+256z^2)^2$

Local exponents

$0$ $\frac{ 17}{ 16}+\frac{ 3}{ 4}\sqrt{ 2}$ $\infty$ $\frac{ 17}{ 16}-\frac{ 3}{ 4}\sqrt{ 2}$ $s_1$ $s_2$
$0$ $0$ $\frac{ 3}{ 2}$ $0$ $-\frac{ 1}{ 2}$ $-\frac{ 1}{ 2}$
$0$ $-\frac{ 1}{ 2}$ $2$ $-\frac{ 1}{ 2}$ $0$ $0$
$0$ $1$ $2$ $1$ $1$ $1$
$0$ $\frac{ 3}{ 2}$ $\frac{ 5}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 17}{ 16}+\frac{ 3}{ 4}\sqrt{ 2}\)	\(\infty\)	\(\frac{ 17}{ 16}-\frac{ 3}{ 4}\sqrt{ 2}\)	\(s_1\)	\(s_2\)
\(0\)	\(0\)	\(\frac{ 3}{ 2}\)	\(0\)	\(-\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 2}\)
\(0\)	\(-\frac{ 1}{ 2}\)	\(2\)	\(-\frac{ 1}{ 2}\)	\(0\)	\(0\)
\(0\)	\(1\)	\(2\)	\(1\)	\(1\)	\(1\)
\(0\)	\(\frac{ 3}{ 2}\)	\(\frac{ 5}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)

Note:

YY-Operator equivalent to AESZ 44=$ A \ast \gamma$

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

3

New Number: 4.3 | AESZ: | Superseeker: -20 5924 | Hash: 4163e7dfeb4b46f62bda072d071020fc

Degree: 4

$\theta^4-2^{2} x\left(112\theta^4+224\theta^3+271\theta^2+159\theta+36\right)+2^{6} x^{2}\left(1432\theta^4+5728\theta^3+10849\theta^2+10242\theta+3888\right)-2^{8} 3^{4} x^{3}(112\theta^2+336\theta+341)(2\theta+3)^2+2^{14} 3^{8} x^{4}(\theta+2)^2(2\theta+3)(2\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 144, 13320, 432320, -127603800, ...
--> OEIS
Normalized instanton numbers (n₀=1): -20, 199, 5924, 82010, -1170848, ... ; Common denominator:...

Discriminant

$(1-224z+20736z^2)^2$

Local exponents

$0$ $\frac{ 7}{ 1296}-\frac{ 1}{ 324}\sqrt{ 2}I$ $\infty$ $\frac{ 7}{ 1296}+\frac{ 1}{ 324}\sqrt{ 2}I$ $s_1$ $s_2$
$0$ $0$ $\frac{ 3}{ 2}$ $0$ $-\frac{ 1}{ 2}$ $-\frac{ 1}{ 2}$
$0$ $-\frac{ 1}{ 2}$ $2$ $-\frac{ 1}{ 2}$ $0$ $0$
$0$ $1$ $2$ $1$ $1$ $1$
$0$ $\frac{ 3}{ 2}$ $\frac{ 5}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 7}{ 1296}-\frac{ 1}{ 324}\sqrt{ 2}I\)	\(\infty\)	\(\frac{ 7}{ 1296}+\frac{ 1}{ 324}\sqrt{ 2}I\)	\(s_1\)	\(s_2\)
\(0\)	\(0\)	\(\frac{ 3}{ 2}\)	\(0\)	\(-\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 2}\)
\(0\)	\(-\frac{ 1}{ 2}\)	\(2\)	\(-\frac{ 1}{ 2}\)	\(0\)	\(0\)
\(0\)	\(1\)	\(2\)	\(1\)	\(1\)	\(1\)
\(0\)	\(\frac{ 3}{ 2}\)	\(\frac{ 5}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)

Note:

YY-Operator equivalent to AESZ 150=$ A \ast \delta $

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

4

New Number: 4.4 | AESZ: | Superseeker: -48 -32368 | Hash: a0903e578f379289d79849a566639775

Degree: 4

$\theta^4-2^{4} x\left(48\theta^4+96\theta^3+115\theta^2+67\theta+15\right)+2^{9} x^{2}\left(304\theta^4+1216\theta^3+1890\theta^2+1348\theta+375\right)-2^{14} x^{3}(48\theta^2+144\theta+145)(2\theta+3)^2+2^{22} x^{4}(\theta+2)^2(2\theta+3)(2\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 240, 69840, 22068480, 7268490000, ...
--> OEIS
Normalized instanton numbers (n₀=1): -48, -910, -32368, -1409193, -71439120, ... ; Common denominator:...

Discriminant

$(1-384z+4096z^2)^2$

Local exponents

$0$ $\frac{ 3}{ 64}+\frac{ 1}{ 32}\sqrt{ 2}$ $\frac{ 3}{ 64}-\frac{ 1}{ 32}\sqrt{ 2}$ $\infty$ $s_1$ $s_2$
$0$ $0$ $0$ $\frac{ 3}{ 2}$ $-\frac{ 1}{ 2}$ $-\frac{ 1}{ 2}$
$0$ $-\frac{ 1}{ 2}$ $-\frac{ 1}{ 2}$ $2$ $0$ $0$
$0$ $1$ $1$ $2$ $1$ $1$
$0$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 5}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 3}{ 64}+\frac{ 1}{ 32}\sqrt{ 2}\)	\(\frac{ 3}{ 64}-\frac{ 1}{ 32}\sqrt{ 2}\)	\(\infty\)	\(s_1\)	\(s_2\)
\(0\)	\(0\)	\(0\)	\(\frac{ 3}{ 2}\)	\(-\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 2}\)
\(0\)	\(-\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 2}\)	\(2\)	\(0\)	\(0\)
\(0\)	\(1\)	\(1\)	\(2\)	\(1\)	\(1\)
\(0\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 5}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)

Note:

YY-Operator equivalent to $d \ast e \tilde A \ast \epsilon$

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

5

New Number: 4.5 | AESZ: | Superseeker: -36 -62596/3 | Hash: f5b4785eb6dd46eea771050179115d33

Degree: 4

$\theta^4-2^{2} 3 x\left(48\theta^4+96\theta^3+115\theta^2+67\theta+15\right)+2^{4} 3^{2} x^{2}\left(480\theta^4+1920\theta^3+2580\theta^2+1320\theta+151\right)+2^{8} 3^{4} x^{3}(48\theta^2+144\theta+145)(2\theta+3)^2+2^{14} 3^{6} x^{4}(\theta+2)^2(2\theta+3)(2\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 180, 44676, 11798640, 3241596996, ...
--> OEIS
Normalized instanton numbers (n₀=1): -36, -756, -62596/3, -839088, -37432800, ... ; Common denominator:...

Discriminant

$(-1+288z+6912z^2)^2$

Local exponents

$0$ $-\frac{ 1}{ 48}+\frac{ 1}{ 72}\sqrt{ 3}$ $-\frac{ 1}{ 48}-\frac{ 1}{ 72}\sqrt{ 3}$ $\infty$ $s_1$ $s_2$
$0$ $0$ $0$ $\frac{ 3}{ 2}$ $-\frac{ 1}{ 2}$ $-\frac{ 1}{ 2}$
$0$ $-\frac{ 1}{ 2}$ $-\frac{ 1}{ 2}$ $2$ $0$ $0$
$0$ $1$ $1$ $2$ $1$ $1$
$0$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 5}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$

\(0\)	\(-\frac{ 1}{ 48}+\frac{ 1}{ 72}\sqrt{ 3}\)	\(-\frac{ 1}{ 48}-\frac{ 1}{ 72}\sqrt{ 3}\)	\(\infty\)	\(s_1\)	\(s_2\)
\(0\)	\(0\)	\(0\)	\(\frac{ 3}{ 2}\)	\(-\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 2}\)
\(0\)	\(-\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 2}\)	\(2\)	\(0\)	\(0\)
\(0\)	\(1\)	\(1\)	\(2\)	\(1\)	\(1\)
\(0\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 5}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)

Note:

YY-Operator equivalent to $d \ast e \tilde A\st \epsilon$

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

6

New Number: 4.6 | AESZ: | Superseeker: -28 1036 | Hash: e42780ff25b428328423d5eea814a37a

Degree: 4

$\theta^4-2^{2} x\left(176\theta^4+352\theta^3+427\theta^2+251\theta+57\right)+2^{4} x^{2}\left(11744\theta^4+46976\theta^3+84756\theta^2+75560\theta+27275\right)-2^{8} 5^{3} x^{3}(176\theta^2+528\theta+537)(2\theta+3)^2+2^{14} 5^{6} x^{4}(\theta+2)^2(2\theta+3)(2\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 228, 44716, 8258768, 1469227500, ...
--> OEIS
Normalized instanton numbers (n₀=1): -28, -21, 1036, 53976, 1260496, ... ; Common denominator:...

Discriminant

$(1-352z+32000z^2)^2$

Local exponents

$0$ $\frac{ 11}{ 2000}-\frac{ 1}{ 1000}I$ $\infty$ $\frac{ 11}{ 2000}+\frac{ 1}{ 1000}I$ $s_1$ $s_2$
$0$ $0$ $\frac{ 3}{ 2}$ $0$ $-\frac{ 1}{ 2}$ $-\frac{ 1}{ 2}$
$0$ $-\frac{ 1}{ 2}$ $2$ $-\frac{ 1}{ 2}$ $0$ $0$
$0$ $1$ $2$ $1$ $1$ $1$
$0$ $\frac{ 3}{ 2}$ $\frac{ 5}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 11}{ 2000}-\frac{ 1}{ 1000}I\)	\(\infty\)	\(\frac{ 11}{ 2000}+\frac{ 1}{ 1000}I\)	\(s_1\)	\(s_2\)
\(0\)	\(0\)	\(\frac{ 3}{ 2}\)	\(0\)	\(-\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 2}\)
\(0\)	\(-\frac{ 1}{ 2}\)	\(2\)	\(-\frac{ 1}{ 2}\)	\(0\)	\(0\)
\(0\)	\(1\)	\(2\)	\(1\)	\(1\)	\(1\)
\(0\)	\(\frac{ 3}{ 2}\)	\(\frac{ 5}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)

Note:

YY-operator equivalent to AESZ 121 =$b \ast e \tilde A \ast \eta$

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

7

New Number: 13.5 | AESZ: | Superseeker: 224 4999008 | Hash: 6d924b9c12ee7379761d409ee75e42ab

Degree: 13

$\theta^4-2^{4} x\left(80\theta^4+160\theta^3+152\theta^2+72\theta+15\right)+2^{14} x^{2}\left(24\theta^4+240\theta^3+355\theta^2+230\theta+69\right)+2^{20} x^{3}\left(416\theta^4-2400\theta^3-6216\theta^2-4824\theta-1773\right)-2^{28} x^{4}\left(1840\theta^4-544\theta^3-15328\theta^2-15056\theta-6525\right)+2^{38} 3 x^{5}\left(236\theta^4+1040\theta^3-1629\theta^2-2248\theta-1208\right)+2^{47} 3 x^{6}\left(8\theta^4-1512\theta^3+192\theta^2+951\theta+786\right)-2^{53} 3 x^{7}\left(1568\theta^4-7952\theta^3-4278\theta^2-740\theta+1981\right)+2^{60} 3 x^{8}\left(6976\theta^4-6656\theta^3-9268\theta^2-7912\theta-55\right)-2^{70} x^{9}\left(6680\theta^4+8856\theta^3+8397\theta^2+3060\theta+1017\right)+2^{76} x^{10}\left(22672\theta^4+71840\theta^3+113068\theta^2+90072\theta+30483\right)-2^{84} x^{11}\left(12912\theta^4+62592\theta^3+128336\theta^2+126736\theta+49707\right)+2^{93} 7 x^{12}(2\theta+3)(164\theta^3+810\theta^2+1434\theta+891)-2^{102} 7^{2} x^{13}(\theta+2)^2(2\theta+3)(2\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 240, 44304, 7503616, 1459723536, ...
--> OEIS
Normalized instanton numbers (n₀=1): 224, -22712, 4999008, -855952448, 199163179936, ... ; Common denominator:...

Discriminant

$-(65536z^2-256z+1)^2(117440512z^3-196608z^2+1)^2(256z-1)^3$

Local exponents

$0$ $\frac{ 1}{ 512}+\frac{ 1}{ 512}\sqrt{ 3}I$ $\infty$ $\frac{ 1}{ 256}$ ≈$-0.00161$ ≈$0.001642+0.00161I$ $\frac{ 1}{ 512}-\frac{ 1}{ 512}\sqrt{ 3}I$ ≈$0.001642-0.00161I$
$0$ $0$ $\frac{ 3}{ 2}$ $0$ $0$ $0$ $0$ $0$
$0$ $\frac{ 1}{ 2}$ $2$ $0$ $1$ $1$ $\frac{ 1}{ 2}$ $1$
$0$ $\frac{ 1}{ 2}$ $2$ $0$ $3$ $3$ $\frac{ 1}{ 2}$ $3$
$0$ $1$ $\frac{ 5}{ 2}$ $0$ $4$ $4$ $1$ $4$

\(0\)	\(\frac{ 1}{ 512}+\frac{ 1}{ 512}\sqrt{ 3}I\)	\(\infty\)	\(\frac{ 1}{ 256}\)	≈\(-0.00161\)	≈\(0.001642+0.00161I\)	\(\frac{ 1}{ 512}-\frac{ 1}{ 512}\sqrt{ 3}I\)	≈\(0.001642-0.00161I\)
\(0\)	\(0\)	\(\frac{ 3}{ 2}\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)
\(0\)	\(\frac{ 1}{ 2}\)	\(2\)	\(0\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)	\(1\)
\(0\)	\(\frac{ 1}{ 2}\)	\(2\)	\(0\)	\(3\)	\(3\)	\(\frac{ 1}{ 2}\)	\(3\)
\(0\)	\(1\)	\(\frac{ 5}{ 2}\)	\(0\)	\(4\)	\(4\)	\(1\)	\(4\)

Note:

This is operator "13.5" from ...

Calabi-Yau differential operator database v.3

Summary

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:

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