Calabi-Yau differential operator database v.3.0

Summary

You searched for: h3=15

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New Number: 2.6 | AESZ: 24 | Superseeker: 36 41421 | Hash: 5e8f8f32b5e99693a2956e1240b9fdff

Degree: 2

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

Coefficients of the holomorphic solution: 1, 18, 1710, 246960, 43347150, ...
--> OEIS
Normalized instanton numbers (n₀=1): 36, 837, 41421, 2992851, 266362506, ... ; Common denominator:...

Discriminant

$1-297z-729z^2$

Local exponents

$-\frac{ 11}{ 54}-\frac{ 5}{ 54}\sqrt{ 5}$ $0$ $-\frac{ 11}{ 54}+\frac{ 5}{ 54}\sqrt{ 5}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 3}$
$1$ $0$ $1$ $\frac{ 2}{ 3}$
$1$ $0$ $1$ $\frac{ 4}{ 3}$
$2$ $0$ $2$ $\frac{ 5}{ 3}$

\(-\frac{ 11}{ 54}-\frac{ 5}{ 54}\sqrt{ 5}\)	\(0\)	\(-\frac{ 11}{ 54}+\frac{ 5}{ 54}\sqrt{ 5}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 2}{ 3}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 4}{ 3}\)
\(2\)	\(0\)	\(2\)	\(\frac{ 5}{ 3}\)

Note:

Hadamard product B*b
Related to 7.19, 8.18
This operator corresponds to $(Grass(2,5)\vert 1,1,3)_{-150}$ from arXiv:0802.2908

	Gopakumar-Vafa invariants
g=0	540, 12555, 621315, 44892765, 3995437590, 406684089360, 45426958360155, 5432556927598425,...
g=1	0, 0, -1, 13095, 17230617, 6648808835, 1831575868830, 433375127634753,...
g=2	,...

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-150\)	\(66\)	\(15\)	\(8\)

New Number: 5.36 | AESZ: 219 | Superseeker: 166/5 360988/15 | Hash: b7068bb339f61ebd7c591b7be3fe5893

Degree: 5

$5^{2} \theta^4-2 5 x\left(464\theta^4+1036\theta^3+763\theta^2+245\theta+30\right)-2^{2} 3^{2} x^{2}\left(7064\theta^4+22472\theta^3+26699\theta^2+13200\theta+2340\right)-2^{4} 3^{4} x^{3}\left(3440\theta^4+13320\theta^3+18784\theta^2+10665\theta+2070\right)-2^{6} 3^{8} x^{4}(19\theta^2+59\theta+45)(2\theta+1)^2-2^{8} 3^{9} x^{5}(2\theta+1)^2(2\theta+3)^2$

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 972, 109200, 14949900, ...
--> OEIS
Normalized instanton numbers (n₀=1): 166/5, 638, 360988/15, 7222128/5, 524377242/5, ... ; Common denominator:...

Discriminant

$-(16z+1)(3888z^2+216z-1)(5+36z)^2$

Local exponents

$-\frac{ 5}{ 36}$ $-\frac{ 1}{ 16}$ $-\frac{ 1}{ 36}-\frac{ 1}{ 54}\sqrt{ 3}$ $0$ $-\frac{ 1}{ 36}+\frac{ 1}{ 54}\sqrt{ 3}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $1$ $1$ $0$ $1$ $\frac{ 1}{ 2}$
$3$ $1$ $1$ $0$ $1$ $\frac{ 3}{ 2}$
$4$ $2$ $2$ $0$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 5}{ 36}\)	\(-\frac{ 1}{ 16}\)	\(-\frac{ 1}{ 36}-\frac{ 1}{ 54}\sqrt{ 3}\)	\(0\)	\(-\frac{ 1}{ 36}+\frac{ 1}{ 54}\sqrt{ 3}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(\frac{ 1}{ 2}\)
\(3\)	\(1\)	\(1\)	\(0\)	\(1\)	\(\frac{ 3}{ 2}\)
\(4\)	\(2\)	\(2\)	\(0\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.36" from ...

\(\frac{ 11}{ 2}+600\lambda\)	\(-\frac{ 9}{ 4}-300\lambda\)	\(\frac{ 21}{ 20}+140\lambda\)	\(-.196596398-90\lambda\)
\(14\)	\(-6\)	\(\frac{ 49}{ 15}\)	\(-\frac{ 21}{ 20}-140\lambda\)
\(30\)	\(-15\)	\(8\)	\(-\frac{ 9}{ 4}-300\lambda\)
\(60\)	\(-30\)	\(14\)	\(-\frac{ 7}{ 2}-600\lambda\)

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

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

\(1+150\lambda\)	\(0\)	\(\frac{ 55}{ 2}\lambda\)	\(.35225900e-1\)
\(\frac{ 11}{ 4}\)	\(1\)	\(\frac{ 121}{ 240}\)	\(-\frac{ 55}{ 2}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(15\)	\(0\)	\(\frac{ 11}{ 4}\)	\(1-150\lambda\)

\(-150\lambda\)	\(\frac{ 21}{ 4}\)	\(1\)	\(1\)
\(-\frac{ 11}{ 4}\)	\(-\frac{ 15}{ 2}\)	\(-1\)	\(0\)
\(0\)	\(15\)	\(0\)	\(0\)
\(-15\)	\(0\)	\(0\)	\(0\)