Calabi-Yau differential operator database v.3.0

1

New Number: 3.32 | AESZ: | Superseeker: 128 382592 | Hash: 9b39b616939718654c472dbfb37cdd4e

Degree: 3

$\theta^4-2^{4} x(6\theta^2+6\theta-1)(2\theta+1)^2-2^{10} x^{2}(60\theta^2+120\theta+97)(\theta+1)^2-2^{21} x^{3}(\theta+1)^2(\theta+2)^2$

Maple LaTex

Coefficients of the holomorphic solution: 1, -16, 4624, 678656, 238896400, ...
--> OEIS
Normalized instanton numbers (n₀=1): 128, 4084, 382592, 51510860, 8644861312, ... ; Common denominator:...

Discriminant

$-(512z-1)(1+64z)^2$

Local exponents

$-\frac{ 1}{ 64}$ $0$ $\frac{ 1}{ 512}$ $\infty$
$0$ $0$ $0$ $1$
$-\frac{ 1}{ 2}$ $0$ $1$ $1$
$1$ $0$ $1$ $2$
$\frac{ 3}{ 2}$ $0$ $2$ $2$

\(-\frac{ 1}{ 64}\)	\(0\)	\(\frac{ 1}{ 512}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(1\)
\(-\frac{ 1}{ 2}\)	\(0\)	\(1\)	\(1\)
\(1\)	\(0\)	\(1\)	\(2\)
\(\frac{ 3}{ 2}\)	\(0\)	\(2\)	\(2\)

Note:

Operator equivalent to AESZ 220
B-Incarnation:
Double octic:D.O.244

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

2

New Number: 3.9 | AESZ: ~101 | Superseeker: 13 2650 | Hash: a6878d847acf199583e8168a33967174

Degree: 3

$\theta^4-x\left(113\theta^4+226\theta^3+173\theta^2+60\theta+8\right)-2^{3} x^{2}(\theta+1)^2(119\theta^2+238\theta+92)-2^{2} 11^{2} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 336, 19880, 1420720, ...
--> OEIS
Normalized instanton numbers (n₀=1): 13, 128, 2650, 79400, 2921395, ... ; Common denominator:...

Discriminant

$-(121z-1)(4z+1)^2$

Local exponents

$-\frac{ 1}{ 4}$ $0$ $\frac{ 1}{ 121}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$\frac{ 1}{ 2}$ $0$ $1$ $1$
$\frac{ 1}{ 2}$ $0$ $1$ $2$
$1$ $0$ $2$ $\frac{ 5}{ 2}$

\(-\frac{ 1}{ 4}\)	\(0\)	\(\frac{ 1}{ 121}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(\frac{ 1}{ 2}\)	\(0\)	\(1\)	\(1\)
\(\frac{ 1}{ 2}\)	\(0\)	\(1\)	\(2\)
\(1\)	\(0\)	\(2\)	\(\frac{ 5}{ 2}\)

Note:

Operator equivalent to $AESZ 101=$b \ast b$.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-100\)	\(70\)	\(25\)	\(10\)

3

New Number: 4.41 | AESZ: 220 | Superseeker: 128 382592 | Hash: 671a1aa788ead53985e13ad6774d0189

Degree: 4

$\theta^4-2^{4} x\left(20\theta^4+56\theta^3+38\theta^2+10\theta+1\right)-2^{10} x^{2}\left(84\theta^4+240\theta^3+261\theta^2+134\theta+25\right)-2^{16} x^{3}(2\theta+1)^2(23\theta^2+55\theta+39)-2^{23} x^{4}(2\theta+1)^2(2\theta+3)^2$

Maple LaTex

Coefficients of the holomorphic solution: 1, 16, 3600, 851200, 257328400, ...
--> OEIS
Normalized instanton numbers (n₀=1): 128, 4084, 382592, 51510860, 8644861312, ... ; Common denominator:...

Discriminant

$-(512z-1)(64z+1)^3$

Local exponents

$-\frac{ 1}{ 64}$ $0$ $\frac{ 1}{ 512}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$\frac{ 1}{ 2}$ $0$ $1$ $\frac{ 1}{ 2}$
$\frac{ 3}{ 2}$ $0$ $1$ $\frac{ 3}{ 2}$
$2$ $0$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 1}{ 64}\)	\(0\)	\(\frac{ 1}{ 512}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(\frac{ 1}{ 2}\)	\(0\)	\(1\)	\(\frac{ 1}{ 2}\)
\(\frac{ 3}{ 2}\)	\(0\)	\(1\)	\(\frac{ 3}{ 2}\)
\(2\)	\(0\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

Sporadic Operator.
Reducible to 3.32, so not a primary operator.
B-Incarnation: 81111- x 82--11

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-8\)	\(32\)	\(8\)	\(4\)

4

New Number: 13.4 | AESZ: | Superseeker: 128 341632 | Hash: e2189010cb583bd9f4eab25b75b409cf

Degree: 13

\(\theta^4-2^{5} x\left(4\theta^4+34\theta^3+28\theta^2+11\theta+2\right)-2^{8} x^{2}\left(380\theta^4-872\theta^3-2096\theta^2-1252\theta-351\right)+2^{14} x^{3}\left(1572\theta^4+2760\theta^3-6140\theta^2-4788\theta-1727\right)+2^{18} x^{4}\left(112\theta^4-71968\theta^3+30800\theta^2+34304\theta+16775\right)-2^{25} x^{5}\left(25792\theta^4-66000\theta^3+21380\theta^2+29896\theta+17669\right)+2^{30} x^{6}\left(147184\theta^4-74240\theta^3+128248\theta^2+131808\theta+68259\right)-2^{36} x^{7}\left(204848\theta^4+52096\theta^3+180984\theta^2+135280\theta+61687\right)+2^{42} x^{8}\left(149520\theta^4+15104\theta^3-78056\theta^2-161888\theta-66647\right)-2^{49} x^{9}\left(15408\theta^4-100672\theta^3-280440\theta^2-315312\theta-124965\right)-2^{56} x^{10}\left(10256\theta^4+80960\theta^3+191000\theta^2+198384\theta+76409\right)+2^{63} x^{11}\left(4880\theta^4+28864\theta^3+65080\theta^2+65776\theta+24913\right)-2^{73} x^{12}\left(112\theta^4+648\theta^3+1448\theta^2+1450\theta+545\right)+2^{79} x^{13}\left((2\theta+3)^4\right)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 64, 4496, 339968, 27330832, ...
--> OEIS
Normalized instanton numbers (n₀=1): 128, -4796, 341632, -31623118, 3395329408, ... ; Common denominator:...

Discriminant

$(128z-1)(4096z^2+64z-1)(1048576z^3-28672z^2+160z+1)^2(64z-1)^4$

Local exponents

$-\frac{ 1}{ 128}-\frac{ 1}{ 128}\sqrt{ 5}$ ≈$-0.003609$ $0$ $\frac{ 1}{ 128}$ $-\frac{ 1}{ 128}+\frac{ 1}{ 128}\sqrt{ 5}$ ≈$0.015476-0.004977I$ ≈$0.015476+0.004977I$ $\frac{ 1}{ 64}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 3}{ 2}$
$1$ $1$ $0$ $1$ $1$ $1$ $1$ $0$ $\frac{ 3}{ 2}$
$1$ $3$ $0$ $1$ $1$ $3$ $3$ $0$ $\frac{ 3}{ 2}$
$2$ $4$ $0$ $2$ $2$ $4$ $4$ $0$ $\frac{ 3}{ 2}$

\(-\frac{ 1}{ 128}-\frac{ 1}{ 128}\sqrt{ 5}\)	≈\(-0.003609\)	\(0\)	\(\frac{ 1}{ 128}\)	\(-\frac{ 1}{ 128}+\frac{ 1}{ 128}\sqrt{ 5}\)	≈\(0.015476-0.004977I\)	≈\(0.015476+0.004977I\)	\(\frac{ 1}{ 64}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 3}{ 2}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(0\)	\(\frac{ 3}{ 2}\)
\(1\)	\(3\)	\(0\)	\(1\)	\(1\)	\(3\)	\(3\)	\(0\)	\(\frac{ 3}{ 2}\)
\(2\)	\(4\)	\(0\)	\(2\)	\(2\)	\(4\)	\(4\)	\(0\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "13.4" from ...

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

5

New Number: 12.18 | AESZ: | Superseeker: 128 341632 | Hash: 476ad437981e1b49db55e9fb4d6b4187

Degree: 12

$\theta^4-2^{5} x\left(2\theta^4+34\theta^3+28\theta^2+11\theta+2\right)-2^{8} x^{2}\left(396\theta^4-600\theta^3-1872\theta^2-1164\theta-335\right)+2^{16} x^{3}\left(294\theta^4+840\theta^3-1067\theta^2-906\theta-348\right)+2^{18} x^{4}\left(4816\theta^4-58528\theta^3+13728\theta^2+19808\theta+11207\right)-2^{24} x^{5}\left(46768\theta^4-73472\theta^3+29032\theta^2+39984\theta+24131\right)+2^{34} x^{6}\left(6276\theta^4-48\theta^3+6201\theta^2+5739\theta+2758\right)-2^{36} x^{7}\left(104432\theta^4+52864\theta^3+81768\theta^2+43456\theta+17559\right)+2^{43} x^{8}\left(22544\theta^4-18880\theta^3-79912\theta^2-102672\theta-42103\right)+2^{50} x^{9}\left(3568\theta^4+40896\theta^3+100264\theta^2+106320\theta+41431\right)-2^{57} x^{10}\left(3344\theta^4+20032\theta^3+45368\theta^2+46032\theta+17489\right)+2^{68} x^{11}\left(48\theta^4+276\theta^3+616\theta^2+617\theta+232\right)-2^{73} x^{12}\left((2\theta+3)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 64, 4496, 339968, 27330832, ...
--> OEIS
Normalized instanton numbers (n₀=1): 128, -4796, 341632, -31623118, 3395329408, ... ; Common denominator:...

Discriminant

No data for singularities

Note:

This is operator "12.18" 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:

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

Note:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice