Calabi-Yau differential operator database v.3.0

1

New Number: 2.27 | AESZ: 140 | Superseeker: 108 -4945756 | Hash: 74692097f8183c067c2c4b1a5c93387b

Degree: 2

$\theta^4-2^{2} 3 x(6\theta+1)(6\theta+5)(17\theta^2+17\theta+6)+2^{7} 3^{4} x^{2}(6\theta+1)(6\theta+5)(6\theta+7)(6\theta+11)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 360, 582120, 1274232960, 3204505984680, ...
--> OEIS
Normalized instanton numbers (n₀=1): 108, 54135, -4945756, 7925523138, -2434666062240, ... ; Common denominator:...

Discriminant

$(3888z-1)(3456z-1)$

Local exponents

$0$ $\frac{ 1}{ 3888}$ $\frac{ 1}{ 3456}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 6}$
$0$ $1$ $1$ $\frac{ 5}{ 6}$
$0$ $1$ $1$ $\frac{ 7}{ 6}$
$0$ $2$ $2$ $\frac{ 11}{ 6}$

\(0\)	\(\frac{ 1}{ 3888}\)	\(\frac{ 1}{ 3456}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 6}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 5}{ 6}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 7}{ 6}\)
\(0\)	\(2\)	\(2\)	\(\frac{ 11}{ 6}\)

Note:

Hadamard product $D \ast g$

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-676\)	\(132\)	\(12\)	\(13\)

2

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

Degree: 2

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

Maple LaTex

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	,...

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

3

New Number: 5.100 | AESZ: 347 | Superseeker: 15 27140/3 | Hash: f00de20026c099e75b447c475ab287e4

Degree: 5

$\theta^4-3 x\left(213\theta^4+186\theta^3+149\theta^2+56\theta+8\right)+2^{3} 3^{3} x^{2}\left(702\theta^4+1078\theta^3+949\theta^2+392\theta+60\right)-2^{6} 3^{3} x^{3}\left(9277\theta^4+18432\theta^3+16008\theta^2+6000\theta+840\right)+2^{13} 3^{4} 5 x^{4}(2\theta+1)^2(51\theta^2+69\theta+32)-2^{14} 3^{6} 5^{2} x^{5}(2\theta+1)^2(2\theta+3)^2$

Maple LaTex

Coefficients of the holomorphic solution: 1, 24, 1944, 218400, 28488600, ...
--> OEIS
Normalized instanton numbers (n₀=1): 15, 1329/4, 27140/3, 220680, 5952570, ... ; Common denominator:...

Discriminant

$-(192z-1)(1728z^2-207z+1)(-1+120z)^2$

Local exponents

$0$ $\frac{ 23}{ 384}-\frac{ 11}{ 1152}\sqrt{ 33}$ $\frac{ 1}{ 192}$ $\frac{ 1}{ 120}$ $\frac{ 23}{ 384}+\frac{ 11}{ 1152}\sqrt{ 33}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $1$ $1$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $3$ $1$ $\frac{ 3}{ 2}$
$0$ $2$ $2$ $4$ $2$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 23}{ 384}-\frac{ 11}{ 1152}\sqrt{ 33}\)	\(\frac{ 1}{ 192}\)	\(\frac{ 1}{ 120}\)	\(\frac{ 23}{ 384}+\frac{ 11}{ 1152}\sqrt{ 33}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(3\)	\(1\)	\(\frac{ 3}{ 2}\)
\(0\)	\(2\)	\(2\)	\(4\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.100" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-16\)	\(84\)	\(36\)	\(13\)

4

New Number: 5.106 | AESZ: 360 | Superseeker: 3169/17 16293835/17 | Hash: 502b9ea354d34405e6925ab32d7f1cd2

Degree: 5

$17^{2} \theta^4+17 x\left(10622\theta^4-19904\theta^3-13913\theta^2-3961\theta-510\right)+3^{2} x^{2}\left(1596891\theta^4-10821444\theta^3+10580847\theta^2+6358884\theta+1355036\right)-3^{5} x^{3}\left(5472387\theta^4-81131922\theta^3-52565469\theta^2-9898488\theta+1434596\right)+2^{2} 3^{8} 127 x^{4}\left(318018\theta^4+157911\theta^3-445563\theta^2-476706\theta-130792\right)-2^{2} 3^{12} 5 127^{2} x^{5}(5\theta+3)(5\theta+4)(5\theta+6)(5\theta+7)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 30, 414, -73680, -4205250, ...
--> OEIS
Normalized instanton numbers (n₀=1): 3169/17, -723497/68, 16293835/17, -1870341966/17, 251152956621/17, ... ; Common denominator:...

Discriminant

$-(2278125z^3-33831z^2+182z-1)(17+6858z)^2$

Local exponents

$-\frac{ 17}{ 6858}$ $0$ ≈$0.001817-0.005986I$ ≈$0.001817+0.005986I$ ≈$0.011217$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 3}{ 5}$
$1$ $0$ $1$ $1$ $1$ $\frac{ 4}{ 5}$
$3$ $0$ $1$ $1$ $1$ $\frac{ 6}{ 5}$
$4$ $0$ $2$ $2$ $2$ $\frac{ 7}{ 5}$

\(-\frac{ 17}{ 6858}\)	\(0\)	≈\(0.001817-0.005986I\)	≈\(0.001817+0.005986I\)	≈\(0.011217\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 3}{ 5}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 4}{ 5}\)
\(3\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 6}{ 5}\)
\(4\)	\(0\)	\(2\)	\(2\)	\(2\)	\(\frac{ 7}{ 5}\)

Note:

This is operator "5.106" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(880\)	\(20\)	\(68\)	\(13\)

5

New Number: 5.87 | AESZ: 321 | Superseeker: 35/9 3002/9 | Hash: b786027c217dd5d5c5abac7b1ecc570b

Degree: 5

$3^{4} \theta^4-3^{2} x\left(191\theta^4+862\theta^3+683\theta^2+252\theta+36\right)-2^{5} x^{2}\left(7225\theta^4+24835\theta^3+30634\theta^2+16173\theta+3069\right)-2^{8} x^{3}\left(13251\theta^4+35856\theta^3+27641\theta^2+6966\theta+180\right)-2^{12} 5 x^{4}(2\theta+1)(314\theta^3+363\theta^2+68\theta-31)+2^{16} 5^{2} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 4, 132, 4000, 179620, ...
--> OEIS
Normalized instanton numbers (n₀=1): 35/9, 261/4, 3002/9, 126800/9, 1727129/9, ... ; Common denominator:...

Discriminant

$(32z+1)(32z^2-71z+1)(9+80z)^2$

Local exponents

$-\frac{ 9}{ 80}$ $-\frac{ 1}{ 32}$ $0$ $\frac{ 71}{ 64}-\frac{ 17}{ 64}\sqrt{ 17}$ $\frac{ 71}{ 64}+\frac{ 17}{ 64}\sqrt{ 17}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $1$ $0$ $1$ $1$ $1$
$3$ $1$ $0$ $1$ $1$ $1$
$4$ $2$ $0$ $2$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 9}{ 80}\)	\(-\frac{ 1}{ 32}\)	\(0\)	\(\frac{ 71}{ 64}-\frac{ 17}{ 64}\sqrt{ 17}\)	\(\frac{ 71}{ 64}+\frac{ 17}{ 64}\sqrt{ 17}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(3\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(4\)	\(2\)	\(0\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.87" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-128\)	\(84\)	\(36\)	\(13\)

6

New Number: 8.7 | AESZ: 106 | Superseeker: 12 356 | Hash: fe1c90929d18b81637eaaa93366409ed

Degree: 8

$\theta^4-2^{2} x(3\theta^2+3\theta+1)(11\theta^2+11\theta+3)+2^{4} x^{2}\left(241\theta^4+940\theta^3+1303\theta^2+726\theta+145\right)-2^{7} x^{3}\left(33\theta^4-198\theta^3-607\theta^2-456\theta-117\right)+2^{10} x^{4}\left(239\theta^4+478\theta^3-322\theta^2-561\theta-169\right)+2^{12} x^{5}\left(33\theta^4+330\theta^3+185\theta^2-32\theta-37\right)+2^{14} x^{6}\left(241\theta^4+24\theta^3-71\theta^2+24\theta+23\right)+2^{17} x^{7}(3\theta^2+3\theta+1)(11\theta^2+11\theta+3)+2^{20} x^{8}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 380, 16464, 845676, ...
--> OEIS
Normalized instanton numbers (n₀=1): 12, 20, 356, 34561/4, 161840, ... ; Common denominator:...

Discriminant

$(64z^2+88z-1)(16z^2+44z-1)(1+32z^2)^2$

Local exponents

$-\frac{ 11}{ 8}-\frac{ 5}{ 8}\sqrt{ 5}$ $-\frac{ 11}{ 16}-\frac{ 5}{ 16}\sqrt{ 5}$ $0-\frac{ 1}{ 8}\sqrt{ 2}I$ $0$ $0+\frac{ 1}{ 8}\sqrt{ 2}I$ $-\frac{ 11}{ 16}+\frac{ 5}{ 16}\sqrt{ 5}$ $-\frac{ 11}{ 8}+\frac{ 5}{ 8}\sqrt{ 5}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $0$ $1$ $1$ $1$ $1$
$1$ $1$ $3$ $0$ $3$ $1$ $1$ $1$
$2$ $2$ $4$ $0$ $4$ $2$ $2$ $1$

\(-\frac{ 11}{ 8}-\frac{ 5}{ 8}\sqrt{ 5}\)	\(-\frac{ 11}{ 16}-\frac{ 5}{ 16}\sqrt{ 5}\)	\(0-\frac{ 1}{ 8}\sqrt{ 2}I\)	\(0\)	\(0+\frac{ 1}{ 8}\sqrt{ 2}I\)	\(-\frac{ 11}{ 16}+\frac{ 5}{ 16}\sqrt{ 5}\)	\(-\frac{ 11}{ 8}+\frac{ 5}{ 8}\sqrt{ 5}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(3\)	\(0\)	\(3\)	\(1\)	\(1\)	\(1\)
\(2\)	\(2\)	\(4\)	\(0\)	\(4\)	\(2\)	\(2\)	\(1\)

Note:

Hadamard product $b\ast d$. This operator has a second MUM-point at infinity with the same instanton numbers.
It can be reduced to an operator of degree 4 with a single MUM-point defined over
$Q(\sqrt{?})$.

\(-5.088262696-2.403669772I\)	\(-9.231233148+8.267057761I\)	\(6.171205313+7.544982974I\)	\(3.227450678-3.068539845I\)
\(-6.270037138+8.225054977I\)	\(9.605336680+17.587574161I\)	\(14.123297626-6.142790642I\)	\(-3.279565479-6.225556868I\)
\(-2.195545276+2.457029703I\)	\(2.297135143+5.799775915I\)	\(5.638341607-1.600871223I\)	\(-.887292619-2.058764958I\)
\(-13.680081028+17.945574495I\)	\(18.775280028+38.372889078I\)	\(30.814467548-13.402452309I\)	\(-6.155415591-13.583033166I\)

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

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:

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:

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:

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

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

\(1.+3.275911176I\)	\(0\)	\(1.501459289I\)	\(.894299503\)
\(\frac{ 11}{ 2}\)	\(1\)	\(\frac{ 121}{ 48}\)	\(-1.501459289I\)
\(0\)	\(0\)	\(1\)	\(0\)
\(12\)	\(0\)	\(\frac{ 11}{ 2}\)	\(1.-3.275911176I\)

\(-\frac{ 409488897}{ 125000000}I\)	\(\frac{ 15}{ 2}\)	\(1\)	\(1\)
\(-\frac{ 11}{ 2}\)	\(-6\)	\(-1\)	\(0\)
\(0\)	\(12\)	\(0\)	\(0\)
\(-12\)	\(0\)	\(0\)	\(0\)

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

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