Calabi-Yau differential operator database v.3.0

1

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

Degree: 2

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

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	,...

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

2

New Number: 4.42 | AESZ: 222 | Superseeker: 69/5 29081/5 | Hash: aad7a72e711c9c463396d319e0bf7603

Degree: 4

$5^{2} \theta^4-5 x\left(407\theta^4+1198\theta^3+909\theta^2+310\theta+40\right)-2^{7} x^{2}\left(2103\theta^4+6999\theta^3+8358\theta^2+4050\theta+680\right)-2^{12} x^{3}\left(1387\theta^4+3840\theta^3+3081\theta^2+960\theta+100\right)-2^{21} x^{4}\left((2\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 504, 36800, 3518200, ...
--> OEIS
Normalized instanton numbers (n₀=1): 69/5, 1383/4, 29081/5, 346080, 72023607/5, ... ; Common denominator:...

Discriminant

$-(8192z^2+107z-1)(5+64z)^2$

Local exponents

$-\frac{ 5}{ 64}$ $-\frac{ 107}{ 16384}-\frac{ 51}{ 16384}\sqrt{ 17}$ $0$ $s_1$ $s_2$ $-\frac{ 107}{ 16384}+\frac{ 51}{ 16384}\sqrt{ 17}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $1$ $0$ $1$ $1$ $1$ $\frac{ 1}{ 2}$
$3$ $1$ $0$ $1$ $1$ $1$ $\frac{ 1}{ 2}$
$4$ $2$ $0$ $2$ $2$ $2$ $\frac{ 1}{ 2}$

\(-\frac{ 5}{ 64}\)	\(-\frac{ 107}{ 16384}-\frac{ 51}{ 16384}\sqrt{ 17}\)	\(0\)	\(s_1\)	\(s_2\)	\(-\frac{ 107}{ 16384}+\frac{ 51}{ 16384}\sqrt{ 17}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(3\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(4\)	\(2\)	\(0\)	\(2\)	\(2\)	\(2\)	\(\frac{ 1}{ 2}\)

Note:

Sporadic Operator. There is a second MUM-point hiding at infinity, corresponding to Operator AESZ225/4.43

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-128\)	\(68\)	\(20\)	\(9\)

3

New Number: 5.44 | AESZ: 240 | Superseeker: 231/13 38037/13 | Hash: 8f46cd6968b3b676e251a9d8635637fc

Degree: 5

$13^{2} \theta^4-13 x\left(1449\theta^4+4050\theta^3+3143\theta^2+1118\theta+156\right)-2^{4} x^{2}\left(22760\theta^4-27112\theta^3-121046\theta^2-82316\theta-17589\right)+2^{8} x^{3}\left(3824\theta^4+39936\theta^3-34292\theta^2-63492\theta-19539\right)-2^{16} 3 x^{4}(2\theta+1)(40\theta^3+684\theta^2+1013\theta+399)-2^{20} 3^{2} x^{5}(2\theta+1)(4\theta+3)(4\theta+5)(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 468, 28560, 2135700, ...
--> OEIS
Normalized instanton numbers (n₀=1): 231/13, 826/13, 38037/13, 786076/13, 32662752/13, ... ; Common denominator:...

Discriminant

$-(128z-1)(128z^2-13z+1)(13+192z)^2$

Local exponents

$-\frac{ 13}{ 192}$ $0$ $\frac{ 1}{ 128}$ $\frac{ 13}{ 256}-\frac{ 7}{ 256}\sqrt{ 7}I$ $\frac{ 13}{ 256}+\frac{ 7}{ 256}\sqrt{ 7}I$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$ $1$ $\frac{ 3}{ 4}$
$3$ $0$ $1$ $1$ $1$ $\frac{ 5}{ 4}$
$4$ $0$ $2$ $2$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 13}{ 192}\)	\(0\)	\(\frac{ 1}{ 128}\)	\(\frac{ 13}{ 256}-\frac{ 7}{ 256}\sqrt{ 7}I\)	\(\frac{ 13}{ 256}+\frac{ 7}{ 256}\sqrt{ 7}I\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 3}{ 4}\)
\(3\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 5}{ 4}\)
\(4\)	\(0\)	\(2\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.44" from ...

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

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:

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

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

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

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