Calabi-Yau differential operator database v.3.0

1

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	,...

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

2

New Number: 5.35 | AESZ: 218 | Superseeker: 138/7 42984/7 | Hash: a76111af659715caf2c4344eedd9d678

Degree: 5

$7^{2} \theta^4-2 3 7 x\left(192\theta^4+396\theta^3+303\theta^2+105\theta+14\right)+2^{2} 3 x^{2}\left(1188\theta^4+11736\theta^3+20431\theta^2+12152\theta+2436\right)+2^{2} 3^{3} x^{3}\left(532\theta^4+504\theta^3-3455\theta^2-3829\theta-1036\right)-2^{4} 3^{4} x^{4}(2\theta+1)(36\theta^3+306\theta^2+421\theta+156)-2^{6} 3^{4} x^{5}(2\theta+1)(3\theta+2)(3\theta+4)(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 612, 48000, 4580100, ...
--> OEIS
Normalized instanton numbers (n₀=1): 138/7, 1506/7, 42984/7, 235596, 78950334/7, ... ; Common denominator:...

Discriminant

$-(1296z^3-864z^2+168z-1)(7+12z)^2$

Local exponents

$-\frac{ 7}{ 12}$ $0$ ≈$0.006145$ ≈$0.330261-0.128447I$ ≈$0.330261+0.128447I$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$ $1$ $\frac{ 2}{ 3}$
$3$ $0$ $1$ $1$ $1$ $\frac{ 4}{ 3}$
$4$ $0$ $2$ $2$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 7}{ 12}\)	\(0\)	≈\(0.006145\)	≈\(0.330261-0.128447I\)	≈\(0.330261+0.128447I\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 2}{ 3}\)
\(3\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 4}{ 3}\)
\(4\)	\(0\)	\(2\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.35" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-100\)	\(66\)	\(21\)	\(9\)

3

New Number: 5.70 | AESZ: 287 | Superseeker: 361/21 120472/21 | Hash: 97932196c46a8712f6dcb11165d698be

Degree: 5

$3^{2} 7^{2} \theta^4-3 7 x\left(3289\theta^4+6098\theta^3+4645\theta^2+1596\theta+210\right)+2^{2} 5 x^{2}\left(7712\theta^4-46168\theta^3-106885\theta^2-67410\theta-13629\right)+2^{4} x^{3}\left(106636\theta^4+493416\theta^3+420211\theta^2+116361\theta+6090\right)-2^{8} 5 x^{4}(2\theta+1)(1916\theta^3+2622\theta^2+1077\theta+91)-2^{12} 5^{2} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 10, 510, 38260, 3473470, ...
--> OEIS
Normalized instanton numbers (n₀=1): 361/21, 4780/21, 120472/21, 1537864/7, 216261320/21, ... ; Common denominator:...

Discriminant

$-(64z^3+800z^2+149z-1)(-21+80z)^2$

Local exponents

≈$-12.310784$ ≈$-0.195701$ $0$ ≈$0.006485$ $\frac{ 21}{ 80}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $1$ $0$ $1$ $1$ $1$
$1$ $1$ $0$ $1$ $3$ $1$
$2$ $2$ $0$ $2$ $4$ $\frac{ 3}{ 2}$

≈\(-12.310784\)	≈\(-0.195701\)	\(0\)	≈\(0.006485\)	\(\frac{ 21}{ 80}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(0\)	\(1\)	\(3\)	\(1\)
\(2\)	\(2\)	\(0\)	\(2\)	\(4\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.70" 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

Discriminant

Local exponents

Note:

Characteristic classes:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

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

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

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