Calabi-Yau differential operator database v.3.0

1

New Number: 3.14 | AESZ: | Superseeker: 444 19050964 | Hash: dc96aa2da269d989ee90c49dab6a9c5a

Degree: 3

$\theta^4-2^{2} x\left(452\theta^4+920\theta^3+633\theta^2+173\theta+17\right)-2^{4} x^{2}(4\theta+3)(3808\theta^3+10504\theta^2+8884\theta+1635)-2^{8} 11^{2} x^{3}(4\theta+1)(4\theta+3)(4\theta+7)(4\theta+9)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 68, 42220, 38866320, 43812369900, ...
--> OEIS
Normalized instanton numbers (n₀=1): 444, 57104, 19050964, 9432910668, 5781274591408, ... ; Common denominator:...

Discriminant

$-(1936z-1)(1+64z)^2$

Local exponents

$-\frac{ 1}{ 64}$ $0$ $\frac{ 1}{ 1936}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 4}$
$\frac{ 1}{ 2}$ $0$ $1$ $\frac{ 3}{ 4}$
$1$ $0$ $1$ $\frac{ 7}{ 4}$
$\frac{ 3}{ 2}$ $0$ $2$ $\frac{ 9}{ 4}$

\(-\frac{ 1}{ 64}\)	\(0\)	\(\frac{ 1}{ 1936}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 4}\)
\(\frac{ 1}{ 2}\)	\(0\)	\(1\)	\(\frac{ 3}{ 4}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 7}{ 4}\)
\(\frac{ 3}{ 2}\)	\(0\)	\(2\)	\(\frac{ 9}{ 4}\)

Note:

This is operator Pi = 3.14 (approx.), equivalent to AESZ 238.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-100\)	\(38\)	\(5\)	\(4\)

2

New Number: 3.9 | AESZ: ~101 | Superseeker: 13 2650 | Hash: a6878d847acf199583e8168a33967174

Degree: 3

$\theta^4-x\left(113\theta^4+226\theta^3+173\theta^2+60\theta+8\right)-2^{3} x^{2}(\theta+1)^2(119\theta^2+238\theta+92)-2^{2} 11^{2} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 336, 19880, 1420720, ...
--> OEIS
Normalized instanton numbers (n₀=1): 13, 128, 2650, 79400, 2921395, ... ; Common denominator:...

Discriminant

$-(121z-1)(4z+1)^2$

Local exponents

$-\frac{ 1}{ 4}$ $0$ $\frac{ 1}{ 121}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 2}$
$\frac{ 1}{ 2}$ $0$ $1$ $1$
$\frac{ 1}{ 2}$ $0$ $1$ $2$
$1$ $0$ $2$ $\frac{ 5}{ 2}$

\(-\frac{ 1}{ 4}\)	\(0\)	\(\frac{ 1}{ 121}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(\frac{ 1}{ 2}\)	\(0\)	\(1\)	\(1\)
\(\frac{ 1}{ 2}\)	\(0\)	\(1\)	\(2\)
\(1\)	\(0\)	\(2\)	\(\frac{ 5}{ 2}\)

Note:

Operator equivalent to $AESZ 101=$b \ast b$.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-100\)	\(70\)	\(25\)	\(10\)

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: 5.29 | AESZ: 208 | Superseeker: 274/7 281388/7 | Hash: f1d6dfa8a5cdcc2513dfca4243565b2f

Degree: 5

$7^{2} \theta^4-2 7 x\left(1056\theta^4+1884\theta^3+1397\theta^2+455\theta+56\right)+2^{2} 3 x^{2}\left(22760\theta^4+13672\theta^3-22537\theta^2-18116\theta-3584\right)-2^{4} x^{3}\left(53312\theta^4-162120\theta^3-195172\theta^2-78561\theta-11130\right)-2^{6} 19 x^{4}(1189\theta^2+2533\theta+1646)(2\theta+1)^2+2^{11} 19^{2} x^{5}(2\theta+1)^2(2\theta+3)^2$

Maple LaTex

Coefficients of the holomorphic solution: 1, 16, 1440, 196000, 32418400, ...
--> OEIS
Normalized instanton numbers (n₀=1): 274/7, 6115/7, 281388/7, 2815228, 1699166270/7, ... ; Common denominator:...

Discriminant

$(4z+1)(512z^2-284z+1)(-7+76z)^2$

Local exponents

$-\frac{ 1}{ 4}$ $0$ $\frac{ 71}{ 256}-\frac{ 17}{ 256}\sqrt{ 17}$ $\frac{ 7}{ 76}$ $\frac{ 71}{ 256}+\frac{ 17}{ 256}\sqrt{ 17}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$ $1$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $3$ $1$ $\frac{ 3}{ 2}$
$2$ $0$ $2$ $4$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 1}{ 4}\)	\(0\)	\(\frac{ 71}{ 256}-\frac{ 17}{ 256}\sqrt{ 17}\)	\(\frac{ 7}{ 76}\)	\(\frac{ 71}{ 256}+\frac{ 17}{ 256}\sqrt{ 17}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(3\)	\(1\)	\(\frac{ 3}{ 2}\)
\(2\)	\(0\)	\(2\)	\(4\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.29" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-100\)	\(8\)	\(14\)	\(3\)

5

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 ...

Calabi-Yau differential operator database v.3

Summary

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

\(-100\lambda\)	\(\frac{ 29}{ 12}\)	\(1\)	\(1\)
\(-\frac{ 19}{ 12}\)	\(-\frac{ 5}{ 2}\)	\(-1\)	\(0\)
\(0\)	\(5\)	\(0\)	\(0\)
\(-5\)	\(0\)	\(0\)	\(0\)