Calabi-Yau differential operator database v.3.0

Summary

You searched for: sol=3087

Your search produced 2 matches

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New Number: 5.58 | AESZ: 266 | Superseeker: -18/5 -642/5 | Hash: 5d46913a13c5fa5fa6a547d8b5646133

Degree: 5

$5^{2} \theta^4-3 5 x\left(27\theta^4+108\theta^3+124\theta^2+70\theta+15\right)-2 3^{2} x^{2}\left(1377\theta^4+4536\theta^3+6507\theta^2+4455\theta+1220\right)+2 3^{5} x^{3}\left(567\theta^4+4860\theta^3+11583\theta^2+10665\theta+3445\right)+3^{8} x^{4}\left(729\theta^4+3888\theta^3+6606\theta^2+4662\theta+1184\right)+3^{15} x^{5}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 9, 171, 3087, 69579, ...
--> OEIS
Normalized instanton numbers (n₀=1): -18/5, 117/10, -642/5, 1197, -76788/5, ... ; Common denominator:...

Discriminant

$(1+27z)(27z+5)^2(27z-1)^2$

Local exponents

$-\frac{ 5}{ 27}$ $-\frac{ 1}{ 27}$ $0$ $\frac{ 1}{ 27}$ $\infty$
$0$ $0$ $0$ $0$ $1$
$1$ $1$ $0$ $0$ $1$
$3$ $1$ $0$ $1$ $1$
$4$ $2$ $0$ $1$ $1$

\(-\frac{ 5}{ 27}\)	\(-\frac{ 1}{ 27}\)	\(0\)	\(\frac{ 1}{ 27}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(0\)	\(0\)	\(1\)
\(3\)	\(1\)	\(0\)	\(1\)	\(1\)
\(4\)	\(2\)	\(0\)	\(1\)	\(1\)

Note:

There is a second MUM-point at infinity, corresponding to
Operator AESZ 267/5.59

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

New Number: 8.8 | AESZ: 161 | Superseeker: 9 -1229/3 | Hash: 641d1de9a6564241575c5db52faef694

Degree: 8

$\theta^4-3 x(3\theta^2+3\theta+1)(11\theta^2+11\theta+3)+3^{2} x^{2}\left(366\theta^4+1428\theta^3+1980\theta^2+1104\theta+221\right)-3^{4} x^{3}\left(33\theta^4-198\theta^3-607\theta^2-456\theta-117\right)+3^{5} x^{4}\left(726\theta^4+1452\theta^3-978\theta^2-1704\theta-515\right)+3^{7} x^{5}\left(33\theta^4+330\theta^3+185\theta^2-32\theta-37\right)+3^{8} x^{6}\left(366\theta^4+36\theta^3-108\theta^2+36\theta+35\right)+3^{10} x^{7}(3\theta^2+3\theta+1)(11\theta^2+11\theta+3)+3^{12} x^{8}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 9, 171, 3087, 11259, ...
--> OEIS
Normalized instanton numbers (n₀=1): 9, -81/4, -1229/3, -4644, -26685, ... ; Common denominator:...

Discriminant

$(729z^4+2673z^3+3240z^2-99z+1)(1+27z^2)^2$

Local exponents

≈$-1.848362$ ≈$-1.848362$ $0-\frac{ 1}{ 9}\sqrt{ 3}I$ $0$ $0+\frac{ 1}{ 9}\sqrt{ 3}I$ ≈$0.015028$ ≈$0.015028$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $0$ $1$ $1$ $1$ $1$
$1$ $1$ $3$ $0$ $3$ $1$ $1$ $1$
$2$ $2$ $4$ $0$ $4$ $2$ $2$ $1$

≈\(-1.848362\)	≈\(-1.848362\)	\(0-\frac{ 1}{ 9}\sqrt{ 3}I\)	\(0\)	\(0+\frac{ 1}{ 9}\sqrt{ 3}I\)	≈\(0.015028\)	≈\(0.015028\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(3\)	\(0\)	\(3\)	\(1\)	\(1\)	\(1\)
\(2\)	\(2\)	\(4\)	\(0\)	\(4\)	\(2\)	\(2\)	\(1\)

Note:

Hadamard product $b \qst f$. 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 $Q(\sqrt{\})$

Calabi-Yau differential operator database v.3

Summary

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