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: 5.39 | AESZ: 224 | Superseeker: 59/5 22503/5 | Hash: ba17e8cb074bba75e7a27206be530698

Degree: 5

$5^{2} \theta^4-5 x\left(1057\theta^4+1058\theta^3+819\theta^2+290\theta+40\right)+2^{5} x^{2}\left(10123\theta^4+11419\theta^3+5838\theta^2+1510\theta+180\right)-2^{8} x^{3}\left(30981\theta^4+46560\theta^3+48211\theta^2+25500\theta+5100\right)+2^{14} 11 x^{4}(2\theta+1)(234\theta^3+591\theta^2+581\theta+202)-2^{20} 11^{2} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 312, 19520, 1475320, ...
--> OEIS
Normalized instanton numbers (n₀=1): 59/5, 186, 22503/5, 718052/5, 29091017/5, ... ; Common denominator:...

Discriminant

$-(128z-1)(128z^2-13z+1)(-5+176z)^2$

Local exponents

$0$ $\frac{ 1}{ 128}$ $\frac{ 5}{ 176}$ $\frac{ 13}{ 256}-\frac{ 7}{ 256}\sqrt{ 7}I$ $\frac{ 13}{ 256}+\frac{ 7}{ 256}\sqrt{ 7}I$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $1$ $1$ $1$
$0$ $1$ $3$ $1$ $1$ $1$
$0$ $2$ $4$ $2$ $2$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 1}{ 128}\)	\(\frac{ 5}{ 176}\)	\(\frac{ 13}{ 256}-\frac{ 7}{ 256}\sqrt{ 7}I\)	\(\frac{ 13}{ 256}+\frac{ 7}{ 256}\sqrt{ 7}I\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(3\)	\(1\)	\(1\)	\(1\)
\(0\)	\(2\)	\(4\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.39" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-72\)	\(56\)	\(20\)	\(8\)

4

New Number: 8.27 | AESZ: 302 | Superseeker: 109/5 16777/5 | Hash: e18ddbe4d66a3648b349130bcf119dc7

Degree: 8

$5^{2} \theta^4-5 x\left(1307\theta^4+1798\theta^3+1429\theta^2+530\theta+80\right)+2^{4} x^{2}\left(36361\theta^4+75163\theta^3+80666\theta^2+43340\theta+9120\right)-2^{6} x^{3}\left(434831\theta^4+949176\theta^3+966865\theta^2+545700\theta+118340\right)+2^{11} x^{4}\left(407863\theta^4+845480\theta^3+654048\theta^2+219839\theta+18634\right)-2^{16} x^{5}\left(245714\theta^4+474860\theta^3+365378\theta^2+71595\theta-11507\right)+2^{21} x^{6}\left(90362\theta^4+153828\theta^3+121478\theta^2+35967\theta+2221\right)-2^{26} 11 x^{7}\left(1517\theta^4+2932\theta^3+2087\theta^2+621\theta+59\right)-2^{31} 11^{2} x^{8}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 16, 664, 41920, 3350776, ...
--> OEIS
Normalized instanton numbers (n₀=1): 109/5, 867/4, 16777/5, 662976/5, 26339071/5, ... ; Common denominator:...

Discriminant

$-(-1+143z+32z^2)(32z-1)^2(2816z^2-136z+5)^2$

Local exponents

$-\frac{ 143}{ 64}-\frac{ 19}{ 64}\sqrt{ 57}$ $0$ $-\frac{ 143}{ 64}+\frac{ 19}{ 64}\sqrt{ 57}$ $\frac{ 17}{ 704}-\frac{ 1}{ 704}\sqrt{ 591}I$ $\frac{ 17}{ 704}+\frac{ 1}{ 704}\sqrt{ 591}I$ $\frac{ 1}{ 32}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $0$ $1$ $1$ $1$ $\frac{ 1}{ 2}$ $1$
$1$ $0$ $1$ $3$ $3$ $\frac{ 1}{ 2}$ $1$
$2$ $0$ $2$ $4$ $4$ $1$ $1$

\(-\frac{ 143}{ 64}-\frac{ 19}{ 64}\sqrt{ 57}\)	\(0\)	\(-\frac{ 143}{ 64}+\frac{ 19}{ 64}\sqrt{ 57}\)	\(\frac{ 17}{ 704}-\frac{ 1}{ 704}\sqrt{ 591}I\)	\(\frac{ 17}{ 704}+\frac{ 1}{ 704}\sqrt{ 591}I\)	\(\frac{ 1}{ 32}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)	\(1\)
\(1\)	\(0\)	\(1\)	\(3\)	\(3\)	\(\frac{ 1}{ 2}\)	\(1\)
\(2\)	\(0\)	\(2\)	\(4\)	\(4\)	\(1\)	\(1\)

Note:

This operator has a second MUM-point at infinity corresponding to operator 8.26

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

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

Discriminant

Local exponents

Note:

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

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

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

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