Calabi-Yau differential operator database v.3.0

1

New Number: 2.56 | AESZ: 185 | Superseeker: 6 608 | Hash: 80506439e4d4fdc41f5b16e246a69fbf

Degree: 2

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

Coefficients of the holomorphic solution: 1, 6, 162, 6180, 284130, ...
--> OEIS
Normalized instanton numbers (n₀=1): 6, 93/2, 608, 11754, 275352, ... ; Common denominator:...

Discriminant

$1-72z-432z^2$

Local exponents

$-\frac{ 1}{ 12}-\frac{ 1}{ 18}\sqrt{ 3}$ $0$ $-\frac{ 1}{ 12}+\frac{ 1}{ 18}\sqrt{ 3}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$
$1$ $0$ $1$ $1$
$2$ $0$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 1}{ 12}-\frac{ 1}{ 18}\sqrt{ 3}\)	\(0\)	\(-\frac{ 1}{ 12}+\frac{ 1}{ 18}\sqrt{ 3}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)
\(1\)	\(0\)	\(1\)	\(1\)
\(2\)	\(0\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

Hadamard product $I \ast \zeta$

	Gopakumar-Vafa invariants
g=0	216, 1674, 21888, 423144, 9912672, 265889088, 7814158560, 245812722408,...
g=1	0, 0, 0, 0, 432, 374244, 47016720, 3817401912,...
g=2	,...

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-120\)	\(84\)	\(36\)	\(13\)

2

New Number: 5.122 | AESZ: | Superseeker: 19 18641/3 | Hash: 7cc1a0411f21ffd93f1a9f6468627432

Degree: 5

$\theta^4+x\left(119\theta^4-194\theta^3-143\theta^2-46\theta-6\right)-2^{2} 3^{2} x^{2}\left(46\theta^4+748\theta^3+379\theta^2+150\theta+27\right)-2^{2} 3^{4} x^{3}\left(2164\theta^4+6264\theta^3+7421\theta^2+4131\theta+846\right)-2^{5} 3^{8} x^{4}(2\theta+1)(76\theta^3+222\theta^2+235\theta+85)-2^{8} 3^{12} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 6, 162, 7620, 334530, ...
--> OEIS
Normalized instanton numbers (n₀=1): 19, -170, 18641/3, -163734, 6446745, ... ; Common denominator:...

Discriminant

$-(81z-1)(1296z^2+56z+1)(1+72z)^2$

Local exponents

$-\frac{ 7}{ 324}-\frac{ 1}{ 81}\sqrt{ 2}I$ $-\frac{ 7}{ 324}+\frac{ 1}{ 81}\sqrt{ 2}I$ $-\frac{ 1}{ 72}$ $0$ $\frac{ 1}{ 81}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $1$ $1$ $0$ $1$ $1$
$1$ $1$ $3$ $0$ $1$ $1$
$2$ $2$ $4$ $0$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 7}{ 324}-\frac{ 1}{ 81}\sqrt{ 2}I\)	\(-\frac{ 7}{ 324}+\frac{ 1}{ 81}\sqrt{ 2}I\)	\(-\frac{ 1}{ 72}\)	\(0\)	\(\frac{ 1}{ 81}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)
\(1\)	\(1\)	\(3\)	\(0\)	\(1\)	\(1\)
\(2\)	\(2\)	\(4\)	\(0\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

B-Incarnation as fibre product 62211- 623--1

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

3

New Number: 5.77 | AESZ: 307 | Superseeker: 69/11 8883/11 | Hash: 3a2dcd4c59d8fa5b7c57250efeecba62

Degree: 5

$11^{2} \theta^4-3 11 x\left(361\theta^4+530\theta^3+419\theta^2+154\theta+22\right)+2^{2} x^{2}\left(47008\theta^4+45904\theta^3-3251\theta^2-17094\theta-4851\right)-2^{4} 3 x^{3}\left(31436\theta^4+86856\theta^3+160363\theta^2+122133\theta+30294\right)+2^{9} 3^{2} x^{4}(2\theta+1)(1252\theta^3+5442\theta^2+6767\theta+2625)-2^{14} 3^{6} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 6, 162, 6540, 314370, ...
--> OEIS
Normalized instanton numbers (n₀=1): 69/11, 620/11, 8883/11, 171916/11, 4334406/11, ... ; Common denominator:...

Discriminant

$-(81z-1)(64z^2+1)(-11+96z)^2$

Local exponents

$0-\frac{ 1}{ 8}I$ $0$ $0+\frac{ 1}{ 8}I$ $\frac{ 1}{ 81}$ $\frac{ 11}{ 96}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$ $1$ $1$
$1$ $0$ $1$ $1$ $3$ $1$
$2$ $0$ $2$ $2$ $4$ $\frac{ 3}{ 2}$

\(0-\frac{ 1}{ 8}I\)	\(0\)	\(0+\frac{ 1}{ 8}I\)	\(\frac{ 1}{ 81}\)	\(\frac{ 11}{ 96}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)
\(1\)	\(0\)	\(1\)	\(1\)	\(3\)	\(1\)
\(2\)	\(0\)	\(2\)	\(2\)	\(4\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.77" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-108\)	\(78\)	\(33\)	\(12\)

4

New Number: 11.10 | AESZ: | Superseeker: 307/31 30366/31 | Hash: 4af67003a52ef978f182204bfaff3b67

Degree: 11

\(31^{2} \theta^4-31 x\theta(37\theta^3+3404\theta^2+2167\theta+465)-x^{2}\left(3584242\theta^4+13193680\theta^3+15543050\theta^2+9592175\theta+2490912\right)-3^{2} x^{3}\left(19107317\theta^4+73205086\theta^3+112285993\theta^2+86123611\theta+26445852\right)-3 x^{4}\left(1372729742\theta^4+6047894734\theta^3+11016338393\theta^2+9650491725\theta+3283335324\right)-x^{5}\left(61079790533\theta^4+312026249948\theta^3+649293087145\theta^2+630130831252\theta+231606447564\right)-2 3^{2} x^{6}\left(33534165907\theta^4+196973375042\theta^3+458528416805\theta^2+484791515686\theta+189712671726\right)-3^{2} 7 x^{7}\left(64606565117\theta^4+431259053450\theta^3+1107908854519\theta^2+1261805762830\theta+520567245048\right)-3^{4} 7^{2} x^{8}(\theta+1)(4683541363\theta^3+30431977551\theta^2+68128269606\theta+51768680224)-2^{2} 3^{3} 7^{3} x^{9}(\theta+1)(\theta+2)(1489780280\theta^2+7942046183\theta+10944040794)-2^{2} 3^{4} 7^{4} 53 x^{10}(\theta+3)(\theta+2)(\theta+1)(2336627\theta+7400894)-2^{5} 3^{3} 7^{5} 19 53^{2} 97 x^{11}(\theta+1)(\theta+2)(\theta+3)(\theta+4)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 162, 5472, 282366, ...
--> OEIS
Normalized instanton numbers (n₀=1): 307/31, 1814/31, 30366/31, 639686/31, 17126962/31, ... ; Common denominator:...

Discriminant

$-(8z+1)(679z^2+74z-1)(57z^2+15z+1)(7z+1)^2(2226z^2+555z+31)^2$

Local exponents

$-\frac{ 185}{ 1484}-\frac{ 1}{ 4452}\sqrt{ 32001}$ $-\frac{ 1}{ 7}$ $-\frac{ 5}{ 38}-\frac{ 1}{ 114}\sqrt{ 3}I$ $-\frac{ 5}{ 38}+\frac{ 1}{ 114}\sqrt{ 3}I$ $-\frac{ 1}{ 8}$ $-\frac{ 37}{ 679}-\frac{ 32}{ 679}\sqrt{ 2}$ $-\frac{ 185}{ 1484}+\frac{ 1}{ 4452}\sqrt{ 32001}$ $0$ $-\frac{ 37}{ 679}+\frac{ 32}{ 679}\sqrt{ 2}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $\frac{ 1}{ 2}$ $1$ $1$ $1$ $1$ $1$ $0$ $1$ $2$
$3$ $\frac{ 1}{ 2}$ $1$ $1$ $1$ $1$ $3$ $0$ $1$ $3$
$4$ $1$ $2$ $2$ $2$ $2$ $4$ $0$ $2$ $4$

\(-\frac{ 185}{ 1484}-\frac{ 1}{ 4452}\sqrt{ 32001}\)	\(-\frac{ 1}{ 7}\)	\(-\frac{ 5}{ 38}-\frac{ 1}{ 114}\sqrt{ 3}I\)	\(-\frac{ 5}{ 38}+\frac{ 1}{ 114}\sqrt{ 3}I\)	\(-\frac{ 1}{ 8}\)	\(-\frac{ 37}{ 679}-\frac{ 32}{ 679}\sqrt{ 2}\)	\(-\frac{ 185}{ 1484}+\frac{ 1}{ 4452}\sqrt{ 32001}\)	\(0\)	\(-\frac{ 37}{ 679}+\frac{ 32}{ 679}\sqrt{ 2}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(\frac{ 1}{ 2}\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(2\)
\(3\)	\(\frac{ 1}{ 2}\)	\(1\)	\(1\)	\(1\)	\(1\)	\(3\)	\(0\)	\(1\)	\(3\)
\(4\)	\(1\)	\(2\)	\(2\)	\(2\)	\(2\)	\(4\)	\(0\)	\(2\)	\(4\)

Note:

This is operator "11.10" from ...

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

5

New Number: 8.20 | AESZ: 213 | Superseeker: 118/17 672 | Hash: d430b37f4ca641af0b82cbef83547c51

Degree: 8

$17^{2} \theta^4-2 17 x\left(647\theta^4+1240\theta^3+977\theta^2+357\theta+51\right)-2^{2} x^{2}\left(14437\theta^4+89752\theta^3+147734\theta^2+92123\theta+20400\right)+2^{2} 3 x^{3}\left(21538\theta^4+25680\theta^3-41979\theta^2-56151\theta-17442\right)+2^{3} x^{4}\left(51920\theta^4+166384\theta^3-83149\theta^2-217017\theta-79362\right)-2^{4} 3 x^{5}\left(9360\theta^4-26784\theta^3-43813\theta^2-21965\theta-3496\right)-2^{5} 3 x^{6}\left(10160\theta^4-96\theta^3-10535\theta^2-5385\theta-438\right)-2^{8} 3^{2} x^{7}\left(288\theta^4+864\theta^3+1082\theta^2+641\theta+147\right)-2^{11} 3^{2} x^{8}(4\theta+3)(\theta+1)^2(4\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 6, 162, 6252, 290610, ...
--> OEIS
Normalized instanton numbers (n₀=1): 118/17, 873/17, 672, 447987/34, 5358846/17, ... ; Common denominator:...

Discriminant

$-(4z+1)(32z^3+40z^2+78z-1)(-17+18z+48z^2)^2$

Local exponents

$-\frac{ 3}{ 16}-\frac{ 1}{ 48}\sqrt{ 897}$ ≈$-0.631368-1.433512I$ ≈$-0.631368+1.433512I$ $-\frac{ 1}{ 4}$ $0$ ≈$0.012736$ $-\frac{ 3}{ 16}+\frac{ 1}{ 48}\sqrt{ 897}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 3}{ 4}$
$1$ $1$ $1$ $1$ $0$ $1$ $1$ $1$
$3$ $1$ $1$ $1$ $0$ $1$ $3$ $1$
$4$ $2$ $2$ $2$ $0$ $2$ $4$ $\frac{ 5}{ 4}$

\(8+480\lambda\)	\(-\frac{ 7}{ 2}-240\lambda\)	\(\frac{ 91}{ 72}+\frac{ 260}{ 3}\lambda\)	\(-.302703484-\frac{ 140}{ 3}\lambda\)
\(26\)	\(-12\)	\(\frac{ 169}{ 36}\)	\(-\frac{ 91}{ 72}-\frac{ 260}{ 3}\lambda\)
\(72\)	\(-36\)	\(14\)	\(-\frac{ 7}{ 2}-240\lambda\)
\(144\)	\(-72\)	\(26\)	\(-6-480\lambda\)

Calabi-Yau differential operator database v.3

Summary

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

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

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

\(1+120\lambda\)	\(0\)	\(\frac{ 35}{ 3}\lambda\)	\(.9393573e-2\)
\(\frac{ 7}{ 2}\)	\(1\)	\(\frac{ 49}{ 144}\)	\(-\frac{ 35}{ 3}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(36\)	\(0\)	\(\frac{ 7}{ 2}\)	\(1-120\lambda\)

\(-120\lambda\)	\(\frac{ 19}{ 2}\)	\(1\)	\(1\)
\(-\frac{ 7}{ 2}\)	\(-18\)	\(-1\)	\(0\)
\(0\)	\(36\)	\(0\)	\(0\)
\(-36\)	\(0\)	\(0\)	\(0\)