Calabi-Yau differential operator database v.3.0

1

New Number: 2.10 | AESZ: 70 | Superseeker: 27 18089 | Hash: 3d2adae6eaf26a56c76b8b67d92cc5df

Degree: 2

$\theta^4-3 x(3\theta+1)(3\theta+2)(10\theta^2+10\theta+3)+3^{4} x^{2}(3\theta+1)(3\theta+2)(3\theta+4)(3\theta+5)$

Coefficients of the holomorphic solution: 1, 18, 1350, 156240, 22141350, ...
--> OEIS
Normalized instanton numbers (n₀=1): 27, 432, 18089, 997785, 68438142, ... ; Common denominator:...

Discriminant

$(243z-1)(27z-1)$

Local exponents

$0$ $\frac{ 1}{ 243}$ $\frac{ 1}{ 27}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 3}$
$0$ $1$ $1$ $\frac{ 2}{ 3}$
$0$ $1$ $1$ $\frac{ 4}{ 3}$
$0$ $2$ $2$ $\frac{ 5}{ 3}$

\(0\)	\(\frac{ 1}{ 243}\)	\(\frac{ 1}{ 27}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 2}{ 3}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 4}{ 3}\)
\(0\)	\(2\)	\(2\)	\(\frac{ 5}{ 3}\)

Note:

Hadamard product $B\ast c$.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-156\)	\(72\)	\(18\)	\(9\)

2

New Number: 2.2 | AESZ: 15 | Superseeker: 21 15894 | Hash: c8053e0e9c05ef468263fafd5e3fc764

Degree: 2

$\theta^4-3 x(3\theta+1)(3\theta+2)(7\theta^2+7\theta+2)-2^{3} 3^{2} x^{2}(3\theta+1)(3\theta+2)(3\theta+4)(3\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 900, 94080, 11988900, ...
--> OEIS
Normalized instanton numbers (n₀=1): 21, 480, 15894, 894075, 58703151, ... ; Common denominator:...

Discriminant

$-(27z+1)(216z-1)$

Local exponents

$-\frac{ 1}{ 27}$ $0$ $\frac{ 1}{ 216}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 3}$
$1$ $0$ $1$ $\frac{ 2}{ 3}$
$1$ $0$ $1$ $\frac{ 4}{ 3}$
$2$ $0$ $2$ $\frac{ 5}{ 3}$

\(-\frac{ 1}{ 27}\)	\(0\)	\(\frac{ 1}{ 216}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 2}{ 3}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 4}{ 3}\)
\(2\)	\(0\)	\(2\)	\(\frac{ 5}{ 3}\)

Note:

Hadamard product $B\ast a$.

A-Incarnation: diagonal of (3,3)-intersection in $P^2 \times P^2$

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-162\)	\(72\)	\(18\)	\(9\)

3

New Number: 4.69 | AESZ: 350 | Superseeker: 49 173876/9 | Hash: e6de16eb3758d2ed5687f4b2a2abf36b

Degree: 4

$\theta^4-x\left(24+184\theta+545\theta^2+722\theta^3+289\theta^4\right)+2^{3} 3 x^{2}\left(214\theta^4+2734\theta^3+4861\theta^2+2640\theta+468\right)+2^{6} 3^{2} x^{3}\left(1391\theta^4+5184\theta^3+4252\theta^2+1296\theta+126\right)+2^{10} 3^{6} x^{4}\left((2\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 24, 1944, 232800, 34133400, ...
--> OEIS
Normalized instanton numbers (n₀=1): 49, 136, 173876/9, 781152, 57087750, ... ; Common denominator:...

Discriminant

$(256z-1)(81z-1)(1+24z)^2$

Local exponents

$-\frac{ 1}{ 24}$ $0$ $\frac{ 1}{ 256}$ $\frac{ 1}{ 81}$ $\infty$
$0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$ $\frac{ 1}{ 2}$
$3$ $0$ $1$ $1$ $\frac{ 1}{ 2}$
$4$ $0$ $2$ $2$ $\frac{ 1}{ 2}$

\(-\frac{ 1}{ 24}\)	\(0\)	\(\frac{ 1}{ 256}\)	\(\frac{ 1}{ 81}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(3\)	\(0\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(4\)	\(0\)	\(2\)	\(2\)	\(\frac{ 1}{ 2}\)

Note:

Sporadic Operator. There is a second MUM-point hiding at infinity, corresponding to Operator AESZ 351/4.70

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-64\)	\(60\)	\(18\)	\(8\)

4

New Number: 5.33 | AESZ: 216 | Superseeker: 9 14201/3 | Hash: af7027bf24acce4fd0ed5b09e575e2a5

Degree: 5

$\theta^4-3 x\theta(2+11\theta+18\theta^2+27\theta^3)-2 3^{3} x^{2}\left(72\theta^4+414\theta^3+603\theta^2+330\theta+64\right)+2^{2} 3^{5} x^{3}\left(93\theta^4-720\theta^2-708\theta-184\right)+2^{3} 3^{7} x^{4}(2\theta+1)(54\theta^3+405\theta^2+544\theta+200)-2^{4} 3^{10} x^{5}(2\theta+1)(3\theta+2)(3\theta+4)(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 216, 7200, 567000, ...
--> OEIS
Normalized instanton numbers (n₀=1): 9, 225, 14201/3, 154800, 6298596, ... ; Common denominator:...

Discriminant

$-(27z+1)(108z-1)(36z+1)(-1+18z)^2$

Local exponents

$-\frac{ 1}{ 27}$ $-\frac{ 1}{ 36}$ $0$ $\frac{ 1}{ 108}$ $\frac{ 1}{ 18}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $1$ $0$ $1$ $1$ $\frac{ 2}{ 3}$
$1$ $1$ $0$ $1$ $3$ $\frac{ 4}{ 3}$
$2$ $2$ $0$ $2$ $4$ $\frac{ 3}{ 2}$

\(-\frac{ 1}{ 27}\)	\(-\frac{ 1}{ 36}\)	\(0\)	\(\frac{ 1}{ 108}\)	\(\frac{ 1}{ 18}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(\frac{ 2}{ 3}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(3\)	\(\frac{ 4}{ 3}\)
\(2\)	\(2\)	\(0\)	\(2\)	\(4\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.33" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-52\)	\(48\)	\(18\)	\(7\)

5

New Number: 5.75 | AESZ: 298 | Superseeker: 205/9 97622/9 | Hash: e52d50673ec5c795512e2bc3e1017b12

Degree: 5

$3^{4} \theta^4-3^{2} x\left(1993\theta^4+3218\theta^3+2437\theta^2+828\theta+108\right)+2^{5} x^{2}\left(17486\theta^4+25184\theta^3+12239\theta^2+2790\theta+297\right)-2^{8} x^{3}\left(23620\theta^4+34776\theta^3+28905\theta^2+12447\theta+2106\right)+2^{15} x^{4}(2\theta+1)(340\theta^3+618\theta^2+455\theta+129)-2^{22} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 708, 63840, 6989220, ...
--> OEIS
Normalized instanton numbers (n₀=1): 205/9, 3206/9, 97622/9, 496806, 254037095/9, ... ; Common denominator:...

Discriminant

$-(z-1)(1024z^2-192z+1)(-9+128z)^2$

Local exponents

$0$ $\frac{ 3}{ 32}-\frac{ 1}{ 16}\sqrt{ 2}$ $\frac{ 9}{ 128}$ $\frac{ 3}{ 32}+\frac{ 1}{ 16}\sqrt{ 2}$ $1$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $1$ $1$ $1$
$0$ $1$ $3$ $1$ $1$ $1$
$0$ $2$ $4$ $2$ $2$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 3}{ 32}-\frac{ 1}{ 16}\sqrt{ 2}\)	\(\frac{ 9}{ 128}\)	\(\frac{ 3}{ 32}+\frac{ 1}{ 16}\sqrt{ 2}\)	\(1\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(3\)	\(1\)	\(1\)	\(1\)
\(0\)	\(2\)	\(4\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.75" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-92\)	\(12\)	\(18\)	\(4\)

6

New Number: 5.76 | AESZ: 306 | Superseeker: 73/3 11119 | Hash: d14307aa38b16c728ee31e5936937c44

Degree: 5

$3^{2} \theta^4-3 x\left(592\theta^4+1100\theta^3+829\theta^2+279\theta+36\right)+x^{2}\left(13801\theta^4+6652\theta^3-18041\theta^2-14904\theta-3312\right)-2 x^{3}\theta(8461\theta^3-29160\theta^2-28365\theta-7236)-2^{2} 3 7 x^{4}\left(513\theta^4+864\theta^3+487\theta^2+64\theta-16\right)-2^{3} 3 7^{2} x^{5}(\theta+1)^2(3\theta+2)(3\theta+4)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 732, 67080, 7456140, ...
--> OEIS
Normalized instanton numbers (n₀=1): 73/3, 2131/6, 11119, 518671, 29749701, ... ; Common denominator:...

Discriminant

$-(z+1)(54z^2+189z-1)(-3+14z)^2$

Local exponents

$-\frac{ 7}{ 4}-\frac{ 11}{ 36}\sqrt{ 33}$ $-1$ $0$ $-\frac{ 7}{ 4}+\frac{ 11}{ 36}\sqrt{ 33}$ $\frac{ 3}{ 14}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 2}{ 3}$
$1$ $1$ $0$ $1$ $1$ $1$
$1$ $1$ $0$ $1$ $3$ $1$
$2$ $2$ $0$ $2$ $4$ $\frac{ 4}{ 3}$

\(-\frac{ 7}{ 4}-\frac{ 11}{ 36}\sqrt{ 33}\)	\(-1\)	\(0\)	\(-\frac{ 7}{ 4}+\frac{ 11}{ 36}\sqrt{ 33}\)	\(\frac{ 3}{ 14}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 2}{ 3}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(0\)	\(1\)	\(3\)	\(1\)
\(2\)	\(2\)	\(0\)	\(2\)	\(4\)	\(\frac{ 4}{ 3}\)

Note:

This is operator "5.76" from ...

\(4+468\lambda\)	\(-1.-.755979502I\)	\(\frac{ 1}{ 2}+78\lambda\)	\(-.71415832e-1-52\lambda\)
\(9\)	\(-2\)	\(\frac{ 3}{ 2}\)	\(-\frac{ 1}{ 2}-78\lambda\)
\(18\)	\(-6\)	\(4\)	\(-1.-.755979502I\)
\(54\)	\(-18\)	\(9\)	\(-2-468\lambda\)

\(4+324\lambda\)	\(-\frac{ 3}{ 2}-162\lambda\)	\(\frac{ 3}{ 4}+81\lambda\)	\(-\frac{ 222}{ 1223}-54\lambda\)
\(9\)	\(-\frac{ 7}{ 2}\)	\(\frac{ 9}{ 4}\)	\(-\frac{ 3}{ 4}-81\lambda\)
\(18\)	\(-9\)	\(\frac{ 11}{ 2}\)	\(-\frac{ 3}{ 2}-162\lambda\)
\(36\)	\(-18\)	\(9\)	\(-2-324\lambda\)

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:

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

\(1.+.755979502I\)	\(0\)	\(26\lambda\)	\(.31750278e-1\)
\(3\)	\(1\)	\(\frac{ 1}{ 2}\)	\(-26\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(18\)	\(0\)	\(3\)	\(1.-.755979502I\)

\(-\frac{ 377989751}{ 500000000}I\)	\(6\)	\(1\)	\(1\)
\(-3\)	\(-9\)	\(-1\)	\(0\)
\(0\)	\(18\)	\(0\)	\(0\)
\(-18\)	\(0\)	\(0\)	\(0\)

\(1+162\lambda\)	\(0\)	\(27\lambda\)	\(.34239575e-1\)
\(3\)	\(1\)	\(\frac{ 1}{ 2}\)	\(-27\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(18\)	\(0\)	\(3\)	\(1-162\lambda\)

\(-162\lambda\)	\(6\)	\(1\)	\(1\)
\(-3\)	\(-9\)	\(-1\)	\(0\)
\(0\)	\(18\)	\(0\)	\(0\)
\(-18\)	\(0\)	\(0\)	\(0\)