Calabi-Yau differential operator database v.3.0

1

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

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

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

Note:

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

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

2

New Number: 4.47 | AESZ: 239 | Superseeker: 1584 171534960 | Hash: 8e610c3437d7f38e552038bc55399495

Degree: 4

$\theta^4+2^{4} 3 x\left(9\theta^4-198\theta^3-131\theta^2-32\theta-3\right)-2^{11} 3^{2} x^{2}\left(486\theta^4+1215\theta^3+81\theta^2-27\theta-5\right)-2^{16} 3^{5} x^{3}\left(891\theta^4+972\theta^3+675\theta^2+216\theta+25\right)-2^{23} 3^{8} x^{4}(3\theta+1)^2(3\theta+2)^2$

Maple LaTex

Coefficients of the holomorphic solution: 1, 144, 147600, 239904000, 479672701200, ...
--> OEIS
Normalized instanton numbers (n₀=1): 1584, -17874, 171534960, 30012731550, 105934107802896, ... ; Common denominator:...

Discriminant

$-(432z+1)(3456z-1)(1+1728z)^2$

Local exponents

$-\frac{ 1}{ 432}$ $-\frac{ 1}{ 1728}$ $0$ $\frac{ 1}{ 3456}$ $\infty$
$0$ $0$ $0$ $0$ $\frac{ 1}{ 3}$
$1$ $1$ $0$ $1$ $\frac{ 1}{ 3}$
$1$ $3$ $0$ $1$ $\frac{ 2}{ 3}$
$2$ $4$ $0$ $2$ $\frac{ 2}{ 3}$

\(-\frac{ 1}{ 432}\)	\(-\frac{ 1}{ 1728}\)	\(0\)	\(\frac{ 1}{ 3456}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(\frac{ 1}{ 3}\)
\(1\)	\(3\)	\(0\)	\(1\)	\(\frac{ 2}{ 3}\)
\(2\)	\(4\)	\(0\)	\(2\)	\(\frac{ 2}{ 3}\)

Note:

Sporadic Operator.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-18\)	\(28\)	\(4\)	\(3\)

3

New Number: 4.65 | AESZ: | Superseeker: 48 -9104 | Hash: 5ec2790b5eda514313634b7aeb0a295c

Degree: 4

$\theta^4-2^{4} x\left(5\theta^4+34\theta^3+25\theta^2+8\theta+1\right)+2^{11} x^{2}\left(5\theta^4+47\theta^3+90\theta^2+47\theta+8\right)+2^{16} x^{3}\left(51\theta^4+192\theta^3+155\theta^2+48\theta+5\right)+2^{23} x^{4}\left((2\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 16, 144, -70400, -9858800, ...
--> OEIS
Normalized instanton numbers (n₀=1): 48, -1298, -9104, 387230, 102374160, ... ; Common denominator:...

Discriminant

$(32768z^2-208z+1)(1+64z)^2$

Local exponents

$-\frac{ 1}{ 64}$ $0$ $s_1$ $s_2$ $\frac{ 13}{ 4096}-\frac{ 7}{ 4096}\sqrt{ 7}I$ $\frac{ 13}{ 4096}+\frac{ 7}{ 4096}\sqrt{ 7}I$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$ $1$ $1$ $\frac{ 1}{ 2}$
$3$ $0$ $1$ $1$ $1$ $1$ $\frac{ 1}{ 2}$
$4$ $0$ $2$ $2$ $2$ $2$ $\frac{ 1}{ 2}$

\(-\frac{ 1}{ 64}\)	\(0\)	\(s_1\)	\(s_2\)	\(\frac{ 13}{ 4096}-\frac{ 7}{ 4096}\sqrt{ 7}I\)	\(\frac{ 13}{ 4096}+\frac{ 7}{ 4096}\sqrt{ 7}I\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(3\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(4\)	\(0\)	\(2\)	\(2\)	\(2\)	\(2\)	\(\frac{ 1}{ 2}\)

Note:

Sporadic Operator. There is a second MUM-point hiding at infinity,
corresponding to Operator AESZ 295/4.64

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

4

New Number: 5.10 | AESZ: 59 | Superseeker: 30/7 124 | Hash: f47563daeb0f7328bd675f13cfb84a55

Degree: 5

$7^{2} \theta^4-2 7 x\left(257\theta^4+520\theta^3+435\theta^2+175\theta+28\right)+2^{2} x^{2}\left(13497\theta^4+55536\theta^3+81222\theta^2+50337\theta+11396\right)-2^{3} x^{3}\left(17201\theta^4+114996\theta^3+248466\theta^2+202629\theta+55412\right)-2^{4} x^{4}\left(5762\theta^4+29668\theta^3+48150\theta^2+31741\theta+7412\right)-2^{5} 3 x^{5}(4\theta+5)(3\theta+2)(3\theta+4)(4\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 144, 3680, 114400, ...
--> OEIS
Normalized instanton numbers (n₀=1): 30/7, 129/14, 124, 72129/56, 130434/7, ... ; Common denominator:...

Discriminant

$-(4z-1)(16z-1)(54z-1)(7+2z)^2$

Local exponents

$-\frac{ 7}{ 2}$ $0$ $\frac{ 1}{ 54}$ $\frac{ 1}{ 16}$ $\frac{ 1}{ 4}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 2}{ 3}$
$1$ $0$ $1$ $1$ $1$ $\frac{ 3}{ 4}$
$3$ $0$ $1$ $1$ $1$ $\frac{ 5}{ 4}$
$4$ $0$ $2$ $2$ $2$ $\frac{ 4}{ 3}$

\(-\frac{ 7}{ 2}\)	\(0\)	\(\frac{ 1}{ 54}\)	\(\frac{ 1}{ 16}\)	\(\frac{ 1}{ 4}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 2}{ 3}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 3}{ 4}\)
\(3\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 5}{ 4}\)
\(4\)	\(0\)	\(2\)	\(2\)	\(2\)	\(\frac{ 4}{ 3}\)

Note:

This is operator "5.10" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-78\)	\(92\)	\(56\)	\(17\)

5

New Number: 8.84 | AESZ: | Superseeker: 1/5 224/5 | Hash: 258fab6f0a4f132fe597fc6f30e54eea

Degree: 8

$5^{2} \theta^4+5 x\theta^2(-1-2\theta+107\theta^2)+2^{2} x^{2}\left(2174\theta^4+5942\theta^3+8569\theta^2+5200\theta+1200\right)+2^{2} 3^{2} x^{3}\left(308\theta^4-4248\theta^3-17051\theta^2-16785\theta-5280\right)-2^{4} 3^{2} x^{4}\left(7060\theta^4+39500\theta^3+69820\theta^2+52851\theta+14688\right)-2^{6} 3^{4} x^{5}\left(881\theta^4+3974\theta^3+8648\theta^2+7983\theta+2581\right)+2^{7} 3^{4} x^{6}\left(1192\theta^4+2376\theta^3-1132\theta^2-4185\theta-1926\right)+2^{8} 3^{6} x^{7}\left(68\theta^4+568\theta^3+1095\theta^2+811\theta+210\right)-2^{12} 3^{8} x^{8}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, -12, 144, 324, ...
--> OEIS
Normalized instanton numbers (n₀=1): 1/5, -6, 224/5, -448/5, -4334/5, ... ; Common denominator:...

Discriminant

$-(9z-1)(576z^3+368z^2+16z+1)(-5-36z+72z^2)^2$

Local exponents

≈$-0.597246$ $\frac{ 1}{ 4}-\frac{ 1}{ 12}\sqrt{ 19}$ ≈$-0.020821-0.049733I$ ≈$-0.020821+0.049733I$ $0$ $\frac{ 1}{ 9}$ $\frac{ 1}{ 4}+\frac{ 1}{ 12}\sqrt{ 19}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $1$ $0$ $1$ $1$ $1$
$1$ $3$ $1$ $1$ $0$ $1$ $3$ $1$
$2$ $4$ $2$ $2$ $0$ $2$ $4$ $1$

≈\(-0.597246\)	\(\frac{ 1}{ 4}-\frac{ 1}{ 12}\sqrt{ 19}\)	≈\(-0.020821-0.049733I\)	≈\(-0.020821+0.049733I\)	\(0\)	\(\frac{ 1}{ 9}\)	\(\frac{ 1}{ 4}+\frac{ 1}{ 12}\sqrt{ 19}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(1\)	\(3\)	\(1\)	\(1\)	\(0\)	\(1\)	\(3\)	\(1\)
\(2\)	\(4\)	\(2\)	\(2\)	\(0\)	\(2\)	\(4\)	\(1\)

Note:

This is operator "8.84" from ...

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

6

New Number: 9.2 | AESZ: | Superseeker: 9/7 49/3 | Hash: 356d4564e48d7a04e815fa223b6ccc46

Degree: 9

$7^{2} \theta^4+7 x\theta(165\theta^3-102\theta^2-65\theta-14)-2^{3} x^{2}\left(920\theta^4+11726\theta^3+15277\theta^2+9478\theta+2352\right)-2^{4} 3^{2} x^{3}\left(4035\theta^4+19554\theta^3+29157\theta^2+20706\theta+5761\right)-2^{8} 3^{2} x^{4}\left(4156\theta^4+17951\theta^3+28198\theta^2+21045\theta+6096\right)-2^{11} 3^{3} x^{5}\left(1538\theta^4+6560\theta^3+10755\theta^2+8234\theta+2420\right)-2^{13} 3^{4} x^{6}\left(695\theta^4+3051\theta^3+5285\theta^2+4191\theta+1259\right)-2^{14} 3^{5} x^{7}\left(385\theta^4+1802\theta^3+3319\theta^2+2754\theta+855\right)-2^{18} 3^{6} x^{8}(\theta+1)^2(15\theta^2+48\theta+43)-2^{20} 3^{7} x^{9}(\theta+1)^2(\theta+2)^2$

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 24, 144, 3240, ...
--> OEIS
Normalized instanton numbers (n₀=1): 9/7, 47/7, 49/3, 1370/7, 10063/7, ... ; Common denominator:...

Discriminant

$-(8z+1)(24z-1)(3z+1)(4z+1)(12z+1)(7+72z+288z^2)^2$

Local exponents

$-\frac{ 1}{ 3}$ $-\frac{ 1}{ 4}$ $-\frac{ 1}{ 8}-\frac{ 1}{ 24}\sqrt{ 5}I$ $-\frac{ 1}{ 8}$ $-\frac{ 1}{ 8}+\frac{ 1}{ 24}\sqrt{ 5}I$ $-\frac{ 1}{ 12}$ $0$ $\frac{ 1}{ 24}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $1$ $1$ $1$ $0$ $1$ $1$
$1$ $1$ $3$ $1$ $3$ $1$ $0$ $1$ $2$
$2$ $2$ $4$ $2$ $4$ $2$ $0$ $2$ $2$

\(-\frac{ 1}{ 3}\)	\(-\frac{ 1}{ 4}\)	\(-\frac{ 1}{ 8}-\frac{ 1}{ 24}\sqrt{ 5}I\)	\(-\frac{ 1}{ 8}\)	\(-\frac{ 1}{ 8}+\frac{ 1}{ 24}\sqrt{ 5}I\)	\(-\frac{ 1}{ 12}\)	\(0\)	\(\frac{ 1}{ 24}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)
\(1\)	\(1\)	\(3\)	\(1\)	\(3\)	\(1\)	\(0\)	\(1\)	\(2\)
\(2\)	\(2\)	\(4\)	\(2\)	\(4\)	\(2\)	\(0\)	\(2\)	\(2\)

Note:

This is operator "9.2" from ...

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

7

New Number: 1.10 | AESZ: 10 | Superseeker: 928 170869536 | Hash: 51f8135aba94201bd0bbe9b2287a92d5

Degree: 1

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 144, 176400, 341510400, 811620810000, ...
--> OEIS
Normalized instanton numbers (n₀=1): 928, 245616, 170869536, 174999877936, 221984814405088, ... ; Common denominator:...

Discriminant

No data for singularities

Note:

A-incarnation: X(4,4) in P^5(1,1,1,1,2,2)

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:

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

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

Note:

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice