Calabi-Yau differential operator database v.3.0

1

New Number: 2.14 | AESZ: 48 | Superseeker: 24 5832 | Hash: 8081a3989d09a7d612953dac3341d90c

Degree: 2

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 24, 1800, 188160, 23423400, ...
--> OEIS
Normalized instanton numbers (n₀=1): 24, 291/2, 5832, 247116, 12634944, ... ; Common denominator:...

Discriminant

$(216z-1)(108z-1)$

Local exponents

$0$ $\frac{ 1}{ 216}$ $\frac{ 1}{ 108}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 3}$
$0$ $1$ $1$ $\frac{ 2}{ 3}$
$0$ $1$ $1$ $\frac{ 4}{ 3}$
$0$ $2$ $2$ $\frac{ 5}{ 3}$

\(0\)	\(\frac{ 1}{ 216}\)	\(\frac{ 1}{ 108}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 2}{ 3}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 4}{ 3}\)
\(0\)	\(2\)	\(2\)	\(\frac{ 5}{ 3}\)

Note:

B*d

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-162\)	\(84\)	\(24\)	\(11\)

2

New Number: 2.61 | AESZ: 26 | Superseeker: 10 1724 | Hash: f3fc09474973b19b8bdb783e3322eb65

Degree: 2

$\theta^4-2 x(2\theta+1)^2(13\theta^2+13\theta+4)-2^{2} 3 x^{2}(2\theta+1)(3\theta+2)(3\theta+4)(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 288, 15200, 968800, ...
--> OEIS
Normalized instanton numbers (n₀=1): 10, 191/2, 1724, 45680, 1478214, ... ; Common denominator:...

Discriminant

$-(4z+1)(108z-1)$

Local exponents

$-\frac{ 1}{ 4}$ $0$ $\frac{ 1}{ 108}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $\frac{ 2}{ 3}$
$1$ $0$ $1$ $\frac{ 4}{ 3}$
$2$ $0$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 1}{ 4}\)	\(0\)	\(\frac{ 1}{ 108}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 2}{ 3}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 4}{ 3}\)
\(2\)	\(0\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

A-incarnation: $X(1,1,1,1,2) \subset Grass(2,6)$

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-116\)	\(76\)	\(28\)	\(11\)

3

New Number: 5.24 | AESZ: 195 | Superseeker: 285/29 40626/29 | Hash: 49a600431b3e9aaa9d9d6947f8df7d2b

Degree: 5

$29^{2} \theta^4-29 x\left(3026\theta^4+5848\theta^3+4577\theta^2+1653\theta+232\right)+x^{2}\left(5568+57768\theta+239159\theta^2+424220\theta^3+258647\theta^4\right)-x^{3}\left(76560+336864\theta+581647\theta^2+532614\theta^3+272743\theta^4\right)+2^{2} 17 x^{4}\left(1922\theta^4+6193\theta^3+8121\theta^2+4894\theta+1112\right)-2^{2} 3 17^{2} x^{5}(\theta+1)^2(3\theta+2)(3\theta+4)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 264, 13040, 778840, ...
--> OEIS
Normalized instanton numbers (n₀=1): 285/29, 2362/29, 40626/29, 997476/29, 30096841/29, ... ; Common denominator:...

Discriminant

$-(27z^3-67z^2+102z-1)(-29+34z)^2$

Local exponents

$0$ ≈$0.009868$ $\frac{ 29}{ 34}$ ≈$1.235807-1.492036I$ ≈$1.235807+1.492036I$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 2}{ 3}$
$0$ $1$ $1$ $1$ $1$ $1$
$0$ $1$ $3$ $1$ $1$ $1$
$0$ $2$ $4$ $2$ $2$ $\frac{ 4}{ 3}$

\(0\)	≈\(0.009868\)	\(\frac{ 29}{ 34}\)	≈\(1.235807-1.492036I\)	≈\(1.235807+1.492036I\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 2}{ 3}\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(3\)	\(1\)	\(1\)	\(1\)
\(0\)	\(2\)	\(4\)	\(2\)	\(2\)	\(\frac{ 4}{ 3}\)

Note:

This is operator "5.24" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-100\)	\(74\)	\(29\)	\(11\)

4

New Number: 5.40 | AESZ: 226 | Superseeker: 62/5 4060/3 | Hash: 92f95cd33ac4bf18c2d05ce3040c5203

Degree: 5

$5^{2} \theta^4-2 5 x\left(328\theta^4+692\theta^3+551\theta^2+205\theta+30\right)+2^{2} 3 x^{2}\left(5352\theta^4+25416\theta^3+38387\theta^2+23020\theta+4860\right)-2^{4} 3^{3} x^{3}\left(352\theta^4+4520\theta^3+12108\theta^2+10205\theta+2630\right)-2^{6} 3^{3} x^{4}(2\theta+1)(586\theta^3+3039\theta^2+3947\theta+1527)-2^{8} 3^{4} x^{5}(2\theta+1)(6\theta+5)(6\theta+7)(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 396, 19920, 1241100, ...
--> OEIS
Normalized instanton numbers (n₀=1): 62/5, 55, 4060/3, 28790, 861786, ... ; Common denominator:...

Discriminant

$-(16z-1)(108z-1)(12z-1)(5+12z)^2$

Local exponents

$-\frac{ 5}{ 12}$ $0$ $\frac{ 1}{ 108}$ $\frac{ 1}{ 16}$ $\frac{ 1}{ 12}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$ $1$ $\frac{ 5}{ 6}$
$3$ $0$ $1$ $1$ $1$ $\frac{ 7}{ 6}$
$4$ $0$ $2$ $2$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 5}{ 12}\)	\(0\)	\(\frac{ 1}{ 108}\)	\(\frac{ 1}{ 16}\)	\(\frac{ 1}{ 12}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 5}{ 6}\)
\(3\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 7}{ 6}\)
\(4\)	\(0\)	\(2\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.40" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-76\)	\(72\)	\(30\)	\(11\)

5

New Number: 5.78 | AESZ: 308 | Superseeker: 248/29 38708/29 | Hash: 94e96c5d238b2d22a633f4e05ec1ae9f

Degree: 5

$29^{2} \theta^4-2 29 x\left(1318\theta^4+2336\theta^3+1806\theta^2+638\theta+87\right)-2^{2} x^{2}\left(90996\theta^4+744384\theta^3+1267526\theta^2+791584\theta+168345\right)+2^{2} 5^{2} x^{3}\left(34172\theta^4+77256\theta^3-46701\theta^2-110403\theta-36540\right)+2^{4} 5^{4} x^{4}(2\theta+1)(68\theta^3+1842\theta^2+2899\theta+1215)-2^{6} 5^{7} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 6, 210, 9780, 551250, ...
--> OEIS
Normalized instanton numbers (n₀=1): 248/29, 2476/29, 38708/29, 940480/29, 27926248/29, ... ; Common denominator:...

Discriminant

$-(2000z^3+1024z^2+84z-1)(-29+100z)^2$

Local exponents

≈$-0.40534$ ≈$-0.117186$ $0$ ≈$0.010526$ $\frac{ 29}{ 100}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $1$ $0$ $1$ $1$ $1$
$1$ $1$ $0$ $1$ $3$ $1$
$2$ $2$ $0$ $2$ $4$ $\frac{ 3}{ 2}$

≈\(-0.40534\)	≈\(-0.117186\)	\(0\)	≈\(0.010526\)	\(\frac{ 29}{ 100}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(0\)	\(1\)	\(3\)	\(1\)
\(2\)	\(2\)	\(0\)	\(2\)	\(4\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.78" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-104\)	\(74\)	\(29\)	\(11\)

6

New Number: 5.84 | AESZ: 318 | Superseeker: 46/5 1126 | Hash: 3fa38f629ecd5f39b585ce0c1bd88463

Degree: 5

$5^{2} \theta^4-5 x\left(473\theta^4+892\theta^3+696\theta^2+250\theta+35\right)+2 x^{2}\left(1973\theta^4-4636\theta^3-14417\theta^2-10895\theta-2745\right)+2 3^{2} x^{3}\left(343\theta^4+1920\theta^3+1147\theta^2-345\theta-320\right)-3^{4} x^{4}\left(83\theta^4-104\theta^3-458\theta^2-406\theta-114\right)-3^{8} x^{5}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 7, 219, 9961, 546379, ...
--> OEIS
Normalized instanton numbers (n₀=1): 46/5, 717/10, 1126, 51481/2, 3609772/5, ... ; Common denominator:...

Discriminant

$-(z+1)(81z^2+92z-1)(-5+9z)^2$

Local exponents

$-\frac{ 46}{ 81}-\frac{ 13}{ 81}\sqrt{ 13}$ $-1$ $0$ $-\frac{ 46}{ 81}+\frac{ 13}{ 81}\sqrt{ 13}$ $\frac{ 5}{ 9}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $0$ $1$ $1$ $1$
$1$ $1$ $0$ $1$ $3$ $1$
$2$ $2$ $0$ $2$ $4$ $1$

\(-\frac{ 46}{ 81}-\frac{ 13}{ 81}\sqrt{ 13}\)	\(-1\)	\(0\)	\(-\frac{ 46}{ 81}+\frac{ 13}{ 81}\sqrt{ 13}\)	\(\frac{ 5}{ 9}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(0\)	\(1\)	\(3\)	\(1\)
\(2\)	\(2\)	\(0\)	\(2\)	\(4\)	\(1\)

Note:

There is a second MUM-point at infinity,
corresponding to Operator AESZ 319/5.85
Fibre product: 53211- x 632--1(1)

\(\frac{ 13}{ 4}+324\lambda\)	\(-\frac{ 9}{ 16}-81\lambda\)	\(\frac{ 21}{ 64}+\frac{ 189}{ 4}\lambda\)	\(-.54109387e-1-\frac{ 243}{ 8}\lambda\)
\(7+\frac{ 1}{ 500000000}I\)	\(-\frac{ 3}{ 4}+\frac{ 1}{ 1000000000}I\)	\(\frac{ 49}{ 48}-\frac{ 1}{ 1000000000}I\)	\(-\frac{ 21}{ 64}-\frac{ 189}{ 4}\lambda\)
\(12+\frac{ 1}{ 250000000}I\)	\(-3+\frac{ 1}{ 500000000}I\)	\(\frac{ 11}{ 4}-\frac{ 1}{ 500000000}I\)	\(-\frac{ 9}{ 16}-81\lambda\)
\(48.+.19e-7I\)	\(-11.999999983+.80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e-8I\)	\(7-\frac{ 1}{ 125000000}I\)	\(-\frac{ 5}{ 4}-324\lambda\)

\(\frac{ 20}{ 3}+464\lambda\)	\(-\frac{ 17}{ 6}-232\lambda\)	\(1.0456349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349+.414911827I\)	\(-.241563519-.227532292I\)
\(\frac{ 62}{ 3}\)	\(-\frac{ 28}{ 3}\)	\(\frac{ 961}{ 252}\)	\(-1.0456349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349206349-.414911827I\)
\(56\)	\(-28\)	\(\frac{ 34}{ 3}\)	\(-\frac{ 17}{ 6}-232\lambda\)
\(112\)	\(-56\)	\(\frac{ 62}{ 3}\)	\(-\frac{ 14}{ 3}-464\lambda\)

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

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

\(1+162\lambda\)	\(0\)	\(\frac{ 189}{ 8}\lambda\)	\(.25679681e-1\)
\(\frac{ 7}{ 2}\)	\(1\)	\(\frac{ 49}{ 96}\)	\(-\frac{ 189}{ 8}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(24\)	\(0\)	\(\frac{ 7}{ 2}\)	\(1-162\lambda\)

\(-162\lambda\)	\(\frac{ 15}{ 2}\)	\(1\)	\(1\)
\(-\frac{ 7}{ 2}\)	\(-12\)	\(-1\)	\(0\)
\(0\)	\(24\)	\(0\)	\(0\)
\(-24\)	\(0\)	\(0\)	\(0\)

\(1+116\lambda\)	\(0\)	\(.63575199e-1I\)	\(\frac{ 79}{ 7000}\)
\(\frac{ 19}{ 6}\)	\(1\)	\(\frac{ 361}{ 1008}\)	\(-.63575199e-1I\)
\(0\)	\(0\)	\(1\)	\(0\)
\(28\)	\(0\)	\(\frac{ 19}{ 6}\)	\(1-116\lambda\)

\(-116\lambda\)	\(\frac{ 47}{ 6}\)	\(1\)	\(1\)
\(-\frac{ 19}{ 6}\)	\(-14\)	\(-1\)	\(0\)
\(0\)	\(28\)	\(0\)	\(0\)
\(-28\)	\(0\)	\(0\)	\(0\)