Calabi-Yau differential operator database v.3.0

1

New Number: 2.24 | AESZ: 137 | Superseeker: 20 1684/3 | Hash: 198d6c822d6c46225ac2553d60df6539

Degree: 2

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

Coefficients of the holomorphic solution: 1, 24, 1512, 124800, 11730600, ...
--> OEIS
Normalized instanton numbers (n₀=1): 20, 2, 1684/3, 7602, 173472, ... ; Common denominator:...

Discriminant

$(144z-1)(128z-1)$

Local exponents

$0$ $\frac{ 1}{ 144}$ $\frac{ 1}{ 128}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $\frac{ 3}{ 2}$
$0$ $2$ $2$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 1}{ 144}\)	\(\frac{ 1}{ 128}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 3}{ 2}\)
\(0\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

Hadamard product $A \ast g$.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-16\)	\(96\)	\(48\)	\(16\)

2

New Number: 2.52 | AESZ: 16 | Superseeker: 4 644/3 | Hash: 05af0662662bfbec63e3186c4f363313

Degree: 2

$\theta^4-2^{2} x(2\theta+1)^2(5\theta^2+5\theta+2)+2^{8} x^{2}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 168, 5120, 190120, ...
--> OEIS
Normalized instanton numbers (n₀=1): 4, 20, 644/3, 3072, 52512, ... ; Common denominator:...

Discriminant

$(64z-1)(16z-1)$

Local exponents

$0$ $\frac{ 1}{ 64}$ $\frac{ 1}{ 16}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $1$
$0$ $1$ $1$ $1$
$0$ $2$ $2$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 1}{ 64}\)	\(\frac{ 1}{ 16}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)
\(0\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

Hadamard product $I \ast \alpha$
A-Incarnation: diagonal subfamily of 1,1,1,1-intersection in $P^1 \times P^1 \times P^1 \times \P^1$
B-Incarnations:
Fibre products: 62211- x 632--1, S62211

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-128\)	\(96\)	\(48\)	\(16\)

3

New Number: 2.5 | AESZ: 25 | Superseeker: 20 8220 | Hash: 93279abcbeeade30c29508de7784e582

Degree: 2

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 684, 58800, 6129900, ...
--> OEIS
Normalized instanton numbers (n₀=1): 20, 277, 8220, 352994, 18651536, ... ; Common denominator:...

Discriminant

$1-176z-256z^2$

Local exponents

$-\frac{ 11}{ 32}-\frac{ 5}{ 32}\sqrt{ 5}$ $0$ $-\frac{ 11}{ 32}+\frac{ 5}{ 32}\sqrt{ 5}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $\frac{ 3}{ 2}$
$2$ $0$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 11}{ 32}-\frac{ 5}{ 32}\sqrt{ 5}\)	\(0\)	\(-\frac{ 11}{ 32}+\frac{ 5}{ 32}\sqrt{ 5}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 3}{ 2}\)
\(2\)	\(0\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

Hadamard product $A\ast b$

A-incarnation: X(1,2,2) in G(2,5)

	Gopakumar-Vafa invariants
g=0	400, 5540, 164400, 7059880, 373030720, 22532353740, 1493352046000, 105953648564840,...
g=1	0, 0, 0, 1537, 882496, 214941640, 37001766880, 5388182343297,...
g=2	,...

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-120\)	\(68\)	\(20\)	\(9\)

4

New Number: 2.70 | AESZ: | Superseeker: 20 28820/3 | Hash: 336ddc1188eadae2f4b4c470a17f4ec1

Degree: 2

$\theta^4-2^{2} x\left(128\theta^4+352\theta^3+413\theta^2+237\theta+54\right)+2^{7} x^{2}(4\theta+5)(2\theta+3)(8\theta+9)(8\theta+13)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 216, 49896, 11872896, 2872063656, ...
--> OEIS
Normalized instanton numbers (n₀=1): 20, 290, 28820/3, 454190, 26517920, ... ; Common denominator:...

Discriminant

$(256z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 256}$ $\infty$
$0$ $0$ $\frac{ 9}{ 8}$
$0$ $\frac{ 1}{ 4}$ $\frac{ 5}{ 4}$
$0$ $-\frac{ 3}{ 4}$ $\frac{ 3}{ 2}$
$0$ $1$ $\frac{ 13}{ 8}$

\(0\)	\(\frac{ 1}{ 256}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 9}{ 8}\)
\(0\)	\(\frac{ 1}{ 4}\)	\(\frac{ 5}{ 4}\)
\(0\)	\(-\frac{ 3}{ 4}\)	\(\frac{ 3}{ 2}\)
\(0\)	\(1\)	\(\frac{ 13}{ 8}\)

Note:

Operator equivalent to (:aesz 255)

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

5

New Number: 4.72 | AESZ: 361 | Superseeker: 20 -119332/9 | Hash: f55eaa640956f064f5230c04d8173d60

Degree: 4

$\theta^4-2^{2} x\left(80\theta^4+88\theta^3+67\theta^2+23\theta+3\right)+2^{4} 3 x^{2}\left(928\theta^4+2080\theta^3+2176\theta^2+972\theta+153\right)-2^{10} 3^{2} x^{3}\left(272\theta^4+648\theta^3+511\theta^2+162\theta+18\right)+2^{12} 3^{6} x^{4}\left((2\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 324, -6000, -2778300, ...
--> OEIS
Normalized instanton numbers (n₀=1): 20, -139, -119332/9, -462222, -2113440, ... ; Common denominator:...

Discriminant

$(20736z^2-224z+1)(-1+48z)^2$

Local exponents

$0$ $s_1$ $s_2$ $\frac{ 7}{ 1296}-\frac{ 1}{ 324}\sqrt{ 2}I$ $\frac{ 7}{ 1296}+\frac{ 1}{ 324}\sqrt{ 2}I$ $\frac{ 1}{ 48}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $1$ $1$ $1$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $1$ $1$ $3$ $\frac{ 1}{ 2}$
$0$ $2$ $2$ $2$ $2$ $4$ $\frac{ 1}{ 2}$

\(0\)	\(s_1\)	\(s_2\)	\(\frac{ 7}{ 1296}-\frac{ 1}{ 324}\sqrt{ 2}I\)	\(\frac{ 7}{ 1296}+\frac{ 1}{ 324}\sqrt{ 2}I\)	\(\frac{ 1}{ 48}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(3\)	\(\frac{ 1}{ 2}\)
\(0\)	\(2\)	\(2\)	\(2\)	\(2\)	\(4\)	\(\frac{ 1}{ 2}\)

Note:

Sporadic Operator. There is a second MUM-point hiding at infinity, corresponding to Operator AESZ 362/4.73

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

6

New Number: 5.73 | AESZ: 293 | Superseeker: 20 13188 | Hash: f19eeaee48396d15d7cf7be47d7d48a7

Degree: 5

$\theta^4-2^{2} x\left(54\theta^4+66\theta^3+49\theta^2+16\theta+2\right)+2^{4} x^{2}\left(417\theta^4-306\theta^3-1219\theta^2-776\theta-154\right)+2^{8} x^{3}\left(166\theta^4+1920\theta^3+1589\theta^2+432\theta+23\right)-2^{12} 7 x^{4}(2\theta+1)(38\theta^3+45\theta^2+12\theta-2)-2^{14} 7^{2} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 528, 45440, 4763920, ...
--> OEIS
Normalized instanton numbers (n₀=1): 20, 867/2, 13188, 609734, 35512476, ... ; Common denominator:...

Discriminant

$-(16z+1)(256z^2+176z-1)(-1+28z)^2$

Local exponents

$-\frac{ 11}{ 32}-\frac{ 5}{ 32}\sqrt{ 5}$ $-\frac{ 1}{ 16}$ $0$ $-\frac{ 11}{ 32}+\frac{ 5}{ 32}\sqrt{ 5}$ $\frac{ 1}{ 28}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $1$ $0$ $1$ $1$ $1$
$1$ $1$ $0$ $1$ $3$ $1$
$2$ $2$ $0$ $2$ $4$ $\frac{ 3}{ 2}$

\(-\frac{ 11}{ 32}-\frac{ 5}{ 32}\sqrt{ 5}\)	\(-\frac{ 1}{ 16}\)	\(0\)	\(-\frac{ 11}{ 32}+\frac{ 5}{ 32}\sqrt{ 5}\)	\(\frac{ 1}{ 28}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(0\)	\(1\)	\(3\)	\(1\)
\(2\)	\(2\)	\(0\)	\(2\)	\(4\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.73" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-72\)	\(52\)	\(16\)	\(7\)

7

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
$Q(\sqrt{?})$.

\(.924152876+.943952044I\)	\(.188111204+1.275393351I\)	\(.593885779-.164736576I\)	\(-.98788356e-1-.111719004I\)
\(.205812544+3.214663066I\)	\(2.256285526+4.201440782I\)	\(1.918711704-.844672672I\)	\(-.387126151-.327818207I\)
\(-.81202836e-1+5.639775328I\)	\(1.617313859+7.506248400I\)	\(4.465075410-1.211574339I\)	\(-.630856884-.62528948587308939323761000463177396943029180176007410838351088466882816118573413617415470125057897174617878647522000926354793886058360352014821676702176933765632237146827234830940250115794349235757295I\)
\(2.469750531+38.575956789I\)	\(15.075426310+50.417289384I\)	\(23.024540442-10.136072063I\)	\(-3.645513812-3.933818487I\)

\(5+512\lambda\)	\(-1.-.620290874I\)	\(\frac{ 1}{ 3}+\frac{ 128}{ 3}\lambda\)	\(-\frac{ 1}{ 7}7^{ \frac{ 5}{ 8})exp(-\frac{ 9}{ 4}}Zeta(5)^(\frac{ 1}{ 4})-\frac{ 64}{ 3}\lambda\)
\(16\)	\(-3\)	\(\frac{ 4}{ 3}\)	\(-\frac{ 1}{ 3}-\frac{ 128}{ 3}\lambda\)
\(48\)	\(-12\)	\(5\)	\(-1.-.620290874I\)
\(192\)	\(-48\)	\(16\)	\(-3-512\lambda\)

\(\frac{ 16}{ 3}+480\lambda\)	\(-\frac{ 13}{ 6}-240\lambda\)	\(\frac{ 299}{ 360}+92\lambda\)	\(-.167088493-52\lambda\)
\(\frac{ 46}{ 3}\)	\(-\frac{ 20}{ 3}\)	\(\frac{ 529}{ 180}\)	\(-\frac{ 299}{ 360}-92\lambda\)
\(40\)	\(-20\)	\(\frac{ 26}{ 3}\)	\(-\frac{ 13}{ 6}-240\lambda\)
\(80\)	\(-40\)	\(\frac{ 46}{ 3}\)	\(-\frac{ 10}{ 3}-480\lambda\)

\(\frac{ 1}{ 2}-60I\lambda+I(-\frac{ 1}{ 2}-60I\lambda)\)	\(\frac{ 1}{ 12}-\frac{ 1}{ 12}I\)	\(-5I\lambda+5\lambda\)	\(-.20732598e-1+.20732598e-1I\)
\(1-I\)	\(\frac{ 1}{ 2}-\frac{ 1}{ 2}I\)	\(\frac{ 1}{ 3}-\frac{ 1}{ 3}I\)	\(5I\lambda-5\lambda\)
\(0\)	\(-1+I\)	\(\frac{ 1}{ 2}-\frac{ 1}{ 2}I\)	\(\frac{ 1}{ 12}-\frac{ 1}{ 12}I\)
\(12-12I\)	\(0\)	\(1-I\)	\(\frac{ 1}{ 2}+60I\lambda+I(-\frac{ 1}{ 2}+60I\lambda)\)

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

\(1+16\lambda\)	\(0\)	\(\frac{ 4}{ 3}\lambda\)	\(.125248e-3\)
\(4\)	\(1\)	\(\frac{ 1}{ 3}\)	\(-\frac{ 4}{ 3}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(48\)	\(0\)	\(4\)	\(1-16\lambda\)

\(-16\lambda\)	\(12\)	\(1\)	\(1\)
\(-4\)	\(-24\)	\(-1\)	\(0\)
\(0\)	\(48\)	\(0\)	\(0\)
\(-48\)	\(0\)	\(0\)	\(0\)

\(1.+.620290874I\)	\(0\)	\(\frac{ 32}{ 3}\lambda\)	\(.8015849e-2\)
\(4\)	\(1\)	\(\frac{ 1}{ 3}\)	\(-\frac{ 32}{ 3}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(48\)	\(0\)	\(4\)	\(1.-.620290874I\)

\(-\frac{ 310145437}{ 500000000}I\)	\(12\)	\(1\)	\(1\)
\(-4\)	\(-24\)	\(-1\)	\(0\)
\(0\)	\(48\)	\(0\)	\(0\)
\(-48\)	\(0\)	\(0\)	\(0\)

\(1+120\lambda\)	\(0\)	\(17\lambda\)	\(.16908432e-1\)
\(\frac{ 17}{ 6}\)	\(1\)	\(\frac{ 289}{ 720}\)	\(-17\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(20\)	\(0\)	\(\frac{ 17}{ 6}\)	\(1-120\lambda\)

\(-120\lambda\)	\(\frac{ 37}{ 6}\)	\(1\)	\(1\)
\(-\frac{ 17}{ 6}\)	\(-10\)	\(-1\)	\(0\)
\(0\)	\(20\)	\(0\)	\(0\)
\(-20\)	\(0\)	\(0\)	\(0\)

\(60I\lambda+\frac{ 1}{ 2}+\frac{ 1}{ 2}I-60\lambda\)	\(3-3I\)	\(-I\)	\(1\)
\(-1+I\)	\(-6+6I\)	\(I\)	\(0\)
\(0\)	\(12-12I\)	\(0\)	\(0\)
\(-12+12I\)	\(0\)	\(0\)	\(0\)

Calabi-Yau differential operator database v.3

Summary

Discriminant

Local exponents

Note:

Explicit solution

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:

Explicit solution

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

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