Calabi-Yau differential operator database v.3.0

1

New Number: 2.18 | AESZ: 110 | Superseeker: 36 8076 | Hash: 5060b638cac581d5f0f9dd7f40d90e6c

Degree: 2

$\theta^4-2^{2} 3 x(3\theta+1)(3\theta+2)(8\theta^2+8\theta+3)+2^{8} 3^{2} x^{2}(3\theta+1)(3\theta+2)(3\theta+4)(3\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 72, 14760, 3951360, 1198751400, ...
--> OEIS
Normalized instanton numbers (n₀=1): 36, -144, 8076, -57996, 6960672, ... ; Common denominator:...

Discriminant

$(432z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 432}$ $\infty$
$0$ $0$ $\frac{ 1}{ 3}$
$0$ $\frac{ 1}{ 2}$ $\frac{ 2}{ 3}$
$0$ $\frac{ 1}{ 2}$ $\frac{ 4}{ 3}$
$0$ $1$ $\frac{ 5}{ 3}$

\(0\)	\(\frac{ 1}{ 432}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(0\)	\(\frac{ 1}{ 2}\)	\(\frac{ 2}{ 3}\)
\(0\)	\(\frac{ 1}{ 2}\)	\(\frac{ 4}{ 3}\)
\(0\)	\(1\)	\(\frac{ 5}{ 3}\)

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

2

New Number: 2.26 | AESZ: 139 | Superseeker: 44 22500 | Hash: f5d9215987323abcff6ed8709927af5d

Degree: 2

$\theta^4-2^{2} x(4\theta+1)(4\theta+3)(17\theta^2+17\theta+6)+2^{7} 3^{2} x^{2}(4\theta+1)(4\theta+3)(4\theta+5)(4\theta+7)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 72, 17640, 5765760, 2156754600, ...
--> OEIS
Normalized instanton numbers (n₀=1): 44, 607, 22500, 1444678, 128626784, ... ; Common denominator:...

Discriminant

$(576z-1)(512z-1)$

Local exponents

$0$ $\frac{ 1}{ 576}$ $\frac{ 1}{ 512}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 4}$
$0$ $1$ $1$ $\frac{ 3}{ 4}$
$0$ $1$ $1$ $\frac{ 5}{ 4}$
$0$ $2$ $2$ $\frac{ 7}{ 4}$

\(0\)	\(\frac{ 1}{ 576}\)	\(\frac{ 1}{ 512}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 4}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 3}{ 4}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 5}{ 4}\)
\(0\)	\(2\)	\(2\)	\(\frac{ 7}{ 4}\)

Note:

Hadamard product $C \ast g$

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-344\)	\(120\)	\(24\)	\(14\)

3

New Number: 4.37 | AESZ: 206 | Superseeker: 4 284 | Hash: bd5dae321e1369e7fae153775f84a351

Degree: 4

$\theta^4-2^{2} x\theta(\theta+1)(2\theta+1)^2-2^{5} x^{2}(2\theta+1)(2\theta+3)(11\theta^2+22\theta+12)-2^{4} 3 5^{2} x^{3}(2\theta+1)(2\theta+3)^2(2\theta+5)-2^{8} 19 x^{4}(2\theta+1)(2\theta+3)(2\theta+5)(2\theta+7)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 72, 1200, 44520, ...
--> OEIS
Normalized instanton numbers (n₀=1): 4, 27, 284, 4368, 80968, ... ; Common denominator:...

Discriminant

$-(16z+1)(4864z^3+896z^2+32z-1)$

Local exponents

≈$-0.10185-0.013248I$ ≈$-0.10185+0.013248I$ $-\frac{ 1}{ 16}$ $0$ $s_1$ $s_3$ $s_2$ ≈$0.019489$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $1$ $1$ $0$ $1$ $1$ $1$ $1$ $\frac{ 3}{ 2}$
$1$ $1$ $1$ $0$ $1$ $1$ $1$ $1$ $\frac{ 5}{ 2}$
$2$ $2$ $2$ $0$ $2$ $2$ $2$ $2$ $\frac{ 7}{ 2}$

≈\(-0.10185-0.013248I\)	≈\(-0.10185+0.013248I\)	\(-\frac{ 1}{ 16}\)	\(0\)	\(s_1\)	\(s_3\)	\(s_2\)	≈\(0.019489\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 3}{ 2}\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 5}{ 2}\)
\(2\)	\(2\)	\(2\)	\(0\)	\(2\)	\(2\)	\(2\)	\(2\)	\(\frac{ 7}{ 2}\)

Note:

Sporadic Operator.

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

4

New Number: 4.57 | AESZ: 278 | Superseeker: 243 513936 | Hash: b30f6ac0da69cf91ab39089e6bf1ac8c

Degree: 4

$\theta^4-3 x\left(279\theta^4+882\theta^3+641\theta^2+200\theta+24\right)-2 3^{5} x^{2}\left(72\theta^4-1710\theta^3-3665\theta^2-1864\theta-296\right)+2^{2} 3^{9} x^{3}\left(909\theta^4+3888\theta^3+3082\theta^2+918\theta+92\right)+2^{4} 3^{15} x^{4}(3\theta+1)(2\theta+1)^2(3\theta+2)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 72, 18360, 6552000, 2767980600, ...
--> OEIS
Normalized instanton numbers (n₀=1): 243, -3402, 513936, 2470824, 6888345300, ... ; Common denominator:...

Discriminant

$(729z-1)(432z-1)(1+162z)^2$

Local exponents

$-\frac{ 1}{ 162}$ $0$ $\frac{ 1}{ 729}$ $\frac{ 1}{ 432}$ $\infty$
$0$ $0$ $0$ $0$ $\frac{ 1}{ 3}$
$1$ $0$ $1$ $1$ $\frac{ 1}{ 2}$
$3$ $0$ $1$ $1$ $\frac{ 1}{ 2}$
$4$ $0$ $2$ $2$ $\frac{ 2}{ 3}$

\(-\frac{ 1}{ 162}\)	\(0\)	\(\frac{ 1}{ 729}\)	\(\frac{ 1}{ 432}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(3\)	\(0\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(4\)	\(0\)	\(2\)	\(2\)	\(\frac{ 2}{ 3}\)

Note:

Sporadic Operator.

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

5

New Number: 4.68 | AESZ: 337 | Superseeker: 2043/5 88982631/5 | Hash: dc26e94a7c1daba6f627be36c42019b7

Degree: 4

$5^{2} \theta^4-3 5 x\left(3483\theta^4+6102\theta^3+4241\theta^2+1190\theta+120\right)+2^{5} 3^{2} x^{2}\left(31428\theta^4+35559\theta^3+243\theta^2-4320\theta-740\right)-2^{8} 3^{5} x^{3}\left(7371\theta^4+4860\theta^3+2997\theta^2+1080\theta+140\right)+2^{13} 3^{8} x^{4}(3\theta+1)^2(3\theta+2)^2$

Maple LaTex

Coefficients of the holomorphic solution: 1, 72, 41400, 37396800, 41397463800, ...
--> OEIS
Normalized instanton numbers (n₀=1): 2043/5, 279018/5, 88982631/5, 8604708876, 25774859896713/5, ... ; Common denominator:...

Discriminant

$(23328z^2-1917z+1)(-5+432z)^2$

Local exponents

$0$ $s_1$ $s_2$ $\frac{ 71}{ 1728}-\frac{ 17}{ 1728}\sqrt{ 17}$ $\frac{ 5}{ 432}$ $\frac{ 71}{ 1728}+\frac{ 17}{ 1728}\sqrt{ 17}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 3}$
$0$ $1$ $1$ $1$ $1$ $1$ $\frac{ 1}{ 3}$
$0$ $1$ $1$ $1$ $3$ $1$ $\frac{ 2}{ 3}$
$0$ $2$ $2$ $2$ $4$ $2$ $\frac{ 2}{ 3}$

\(0\)	\(s_1\)	\(s_2\)	\(\frac{ 71}{ 1728}-\frac{ 17}{ 1728}\sqrt{ 17}\)	\(\frac{ 5}{ 432}\)	\(\frac{ 71}{ 1728}+\frac{ 17}{ 1728}\sqrt{ 17}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 3}\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 3}\)
\(0\)	\(1\)	\(1\)	\(1\)	\(3\)	\(1\)	\(\frac{ 2}{ 3}\)
\(0\)	\(2\)	\(2\)	\(2\)	\(4\)	\(2\)	\(\frac{ 2}{ 3}\)

Note:

Sporadic Operator.

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

6

New Number: 5.112 | AESZ: 395 | Superseeker: 4 940 | Hash: 2d13c01eaf16983977dfb0325c5f376e

Degree: 5

$\theta^4-2^{2} x\theta(22\theta^3+8\theta^2+5\theta+1)+2^{5} x^{2}\left(34\theta^4-152\theta^3-265\theta^2-163\theta-36\right)+2^{8} x^{3}\left(142\theta^4+600\theta^3+335\theta^2-39\theta-54\right)-2^{11} 3 x^{4}\left(68\theta^4-56\theta^3-295\theta^2-261\theta-72\right)-2^{15} 3^{2} x^{5}(4\theta+3)(\theta+1)^2(4\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 72, 1728, 72360, ...
--> OEIS
Normalized instanton numbers (n₀=1): 4, 60, 940, 19091, 463904, ... ; Common denominator:...

Discriminant

$-(16z+1)(8z+1)(64z-1)(-1+24z)^2$

Local exponents

$-\frac{ 1}{ 8}$ $-\frac{ 1}{ 16}$ $0$ $\frac{ 1}{ 64}$ $\frac{ 1}{ 24}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 3}{ 4}$
$1$ $1$ $0$ $1$ $1$ $1$
$1$ $1$ $0$ $1$ $3$ $1$
$2$ $2$ $0$ $2$ $4$ $\frac{ 5}{ 4}$

\(-\frac{ 1}{ 8}\)	\(-\frac{ 1}{ 16}\)	\(0\)	\(\frac{ 1}{ 64}\)	\(\frac{ 1}{ 24}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 3}{ 4}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(0\)	\(1\)	\(3\)	\(1\)
\(2\)	\(2\)	\(0\)	\(2\)	\(4\)	\(\frac{ 5}{ 4}\)

Note:

This is operator "5.112" from ...

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

7

New Number: 5.132 | AESZ: | Superseeker: 388 3446444 | Hash: 55a5cb23b8c035363ff67bcc0d5fd556

Degree: 5

$\theta^4+2^{2} x\left(92\theta^4-680\theta^3-481\theta^2-141\theta-18\right)-2^{8} 3^{2} x^{2}\left(192\theta^4+456\theta^3-514\theta^2-323\theta-67\right)-2^{14} 3^{4} x^{3}\left(88\theta^4-312\theta^3-248\theta^2-75\theta-5\right)+2^{20} 3^{7} x^{4}(2\theta+1)(8\theta^3+8\theta^2+\theta-1)-2^{26} 3^{8} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 72, 12456, 3202560, 1030375080, ...
--> OEIS
Normalized instanton numbers (n₀=1): 388, -23196, 3446444, -571523888, 119779121440, ... ; Common denominator:...

Discriminant

$-(144z-1)(576z-1)(64z-1)(1+576z)^2$

Local exponents

$-\frac{ 1}{ 576}$ $0$ $\frac{ 1}{ 576}$ $\frac{ 1}{ 144}$ $\frac{ 1}{ 64}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$ $1$ $1$
$3$ $0$ $1$ $1$ $1$ $1$
$4$ $0$ $2$ $2$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 1}{ 576}\)	\(0\)	\(\frac{ 1}{ 576}\)	\(\frac{ 1}{ 144}\)	\(\frac{ 1}{ 64}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)
\(3\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)
\(4\)	\(0\)	\(2\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

B-Incarnation as fibre product 62211- x 236--1

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

8

New Number: 5.34 | AESZ: 217 | Superseeker: 17/7 5095/21 | Hash: e8743aeac19deca699ff90aaef6b8ea7

Degree: 5

$7^{2} \theta^4+7 x\theta(-14-73\theta-118\theta^2+13\theta^3)-2^{3} 3 x^{2}\left(3378\theta^4+13446\theta^3+18869\theta^2+11158\theta+2352\right)-2^{4} 3^{3} x^{3}\left(3628\theta^4+17920\theta^3+31668\theta^2+22596\theta+5383\right)-2^{8} 3^{3} x^{4}(2\theta+1)(572\theta^3+2370\theta^2+2896\theta+1095)-2^{10} 3^{4} x^{5}(2\theta+1)(6\theta+5)(6\theta+7)(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 72, 720, 37800, ...
--> OEIS
Normalized instanton numbers (n₀=1): 17/7, 254/7, 5095/21, 29600/7, 491991/7, ... ; Common denominator:...

Discriminant

$-(16z+1)(27z+1)(48z-1)(7+24z)^2$

Local exponents

$-\frac{ 7}{ 24}$ $-\frac{ 1}{ 16}$ $-\frac{ 1}{ 27}$ $0$ $\frac{ 1}{ 48}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $1$ $1$ $0$ $1$ $\frac{ 5}{ 6}$
$3$ $1$ $1$ $0$ $1$ $\frac{ 7}{ 6}$
$4$ $2$ $2$ $0$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 7}{ 24}\)	\(-\frac{ 1}{ 16}\)	\(-\frac{ 1}{ 27}\)	\(0\)	\(\frac{ 1}{ 48}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(\frac{ 5}{ 6}\)
\(3\)	\(1\)	\(1\)	\(0\)	\(1\)	\(\frac{ 7}{ 6}\)
\(4\)	\(2\)	\(2\)	\(0\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.34" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-112\)	\(84\)	\(42\)	\(14\)

9

New Number: 5.92 | AESZ: 332 | Superseeker: -16/3 208/3 | Hash: f788b099648b78746af9d38e85874401

Degree: 5

$3^{2} \theta^4+2^{2} 3 x\left(67\theta^4+122\theta^3+100\theta^2+39\theta+6\right)+2^{5} x^{2}\left(1172\theta^4+4298\theta^3+5831\theta^2+3315\theta+678\right)+2^{8} x^{3}\left(3021\theta^4+15912\theta^3+29314\theta^2+20925\theta+4926\right)+2^{11} x^{4}(2\theta+1)(826\theta^3+3543\theta^2+4321\theta+1594)+2^{16} x^{5}(2\theta+1)(4\theta+3)(4\theta+5)(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, -8, 72, 640, -51800, ...
--> OEIS
Normalized instanton numbers (n₀=1): -16/3, -257/6, 208/3, 10444/3, -116608/3, ... ; Common denominator:...

Discriminant

$(32z+1)(2048z^2+52z+1)(8z+3)^2$

Local exponents

$-\frac{ 3}{ 8}$ $-\frac{ 1}{ 32}$ $-\frac{ 13}{ 1024}-\frac{ 7}{ 1024}\sqrt{ 7}I$ $-\frac{ 13}{ 1024}+\frac{ 7}{ 1024}\sqrt{ 7}I$ $0$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $1$ $1$ $1$ $0$ $\frac{ 3}{ 4}$
$3$ $1$ $1$ $1$ $0$ $\frac{ 5}{ 4}$
$4$ $2$ $2$ $2$ $0$ $\frac{ 3}{ 2}$

\(-\frac{ 3}{ 8}\)	\(-\frac{ 1}{ 32}\)	\(-\frac{ 13}{ 1024}-\frac{ 7}{ 1024}\sqrt{ 7}I\)	\(-\frac{ 13}{ 1024}+\frac{ 7}{ 1024}\sqrt{ 7}I\)	\(0\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(1\)	\(1\)	\(0\)	\(\frac{ 3}{ 4}\)
\(3\)	\(1\)	\(1\)	\(1\)	\(0\)	\(\frac{ 5}{ 4}\)
\(4\)	\(2\)	\(2\)	\(2\)	\(0\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.92" from ...

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

10

New Number: 11.8 | AESZ: | Superseeker: 6/17 688/17 | Hash: a0a3e346d09b91b8ad96e54854c136ad

Degree: 11

$17^{2} \theta^4-2 3 17 x\theta^2(117\theta^2+2\theta+1)+2^{2} x^{2}\left(8475\theta^4-64176\theta^3-97010\theta^2-63580\theta-16184\right)+2^{2} x^{3}\left(717094\theta^4+1400796\theta^3+1493367\theta^2+893571\theta+254082\right)-2^{4} x^{4}\left(464294\theta^4-1133264\theta^3-1648391\theta^2-1200310\theta-375336\right)-2^{4} x^{5}\left(18282700\theta^4+46995928\theta^3+83098711\theta^2+73517673\theta+25685438\right)-2^{6} 3 x^{6}\left(2709886\theta^4+7353008\theta^3+18175093\theta^2+18787708\theta+5966228\right)+2^{6} x^{7}\left(154368940\theta^4+947965400\theta^3+2363187035\theta^2+2646307981\theta+1071488886\right)+2^{8} x^{8}(\theta+1)(119648213\theta^3+399067803\theta^2+77665606\theta-498465144)-2^{8} 3 x^{9}(\theta+1)(\theta+2)(120410834\theta^2+865960638\theta+1188072247)-2^{10} 3^{2} 107 x^{10}(\theta+3)(\theta+2)(\theta+1)(218683\theta-39394)+2^{11} 3^{3} 5 107^{2} 137 x^{11}(\theta+1)(\theta+2)(\theta+3)(\theta+4)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 14, 72, 1554, ...
--> OEIS
Normalized instanton numbers (n₀=1): 6/17, 83/17, 688/17, 7350/17, 5150, ... ; Common denominator:...

Discriminant

$(10z+1)(6z-1)(1096z^3+228z^2+14z-1)(2z-1)^2(1284z^2+232z-17)^2$

Local exponents

$-\frac{ 29}{ 321}-\frac{ 1}{ 642}\sqrt{ 8821}$ ≈$-0.124082-0.085658I$ ≈$-0.124082+0.085658I$ $-\frac{ 1}{ 10}$ $0$ ≈$0.040135$ $-\frac{ 29}{ 321}+\frac{ 1}{ 642}\sqrt{ 8821}$ $\frac{ 1}{ 6}$ $\frac{ 1}{ 2}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $1$ $0$ $1$ $1$ $1$ $0$ $2$
$3$ $1$ $1$ $1$ $0$ $1$ $3$ $1$ $1$ $3$
$4$ $2$ $2$ $2$ $0$ $2$ $4$ $2$ $1$ $4$

\(-\frac{ 29}{ 321}-\frac{ 1}{ 642}\sqrt{ 8821}\)	≈\(-0.124082-0.085658I\)	≈\(-0.124082+0.085658I\)	\(-\frac{ 1}{ 10}\)	\(0\)	≈\(0.040135\)	\(-\frac{ 29}{ 321}+\frac{ 1}{ 642}\sqrt{ 8821}\)	\(\frac{ 1}{ 6}\)	\(\frac{ 1}{ 2}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(0\)	\(2\)
\(3\)	\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(3\)	\(1\)	\(1\)	\(3\)
\(4\)	\(2\)	\(2\)	\(2\)	\(0\)	\(2\)	\(4\)	\(2\)	\(1\)	\(4\)

Note:

This is operator "11.8" from ...

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

11

New Number: 12.6 | AESZ: | Superseeker: 5 953/3 | Hash: 4ea78627bfc56ef9555d9b6b3c949e7a

Degree: 12

$\theta^4-x\theta(5\theta^3+46\theta^2+29\theta+6)-2 3 x^{2}\left(258\theta^4+1038\theta^3+1387\theta^2+818\theta+192\right)-2^{2} 3^{3} x^{3}\left(381\theta^4+1664\theta^3+2804\theta^2+2126\theta+624\right)-2^{4} 3^{3} x^{4}\left(1231\theta^4+5927\theta^3+11019\theta^2+9266\theta+3000\right)-2^{4} 3^{4} x^{5}\left(2621\theta^4+16730\theta^3+39069\theta^2+35141\theta+11748\right)-2^{5} 3^{5} x^{6}\left(150\theta^4+11268\theta^3+45560\theta^2+50253\theta+18756\right)+2^{6} 3^{7} x^{7}\left(1024\theta^4+800\theta^3-8483\theta^2-13641\theta-6108\right)+2^{8} 3^{7} x^{8}\left(1724\theta^4+6608\theta^3+1047\theta^2-7027\theta-4488\right)+2^{11} 3^{8} x^{9}\left(74\theta^4+1416\theta^3+1889\theta^2+687\theta-81\right)-2^{13} 3^{10} x^{10}\left(26\theta^4-16\theta^3-125\theta^2-128\theta-39\right)-2^{14} 3^{11} x^{11}(\theta+1)(16\theta^3+40\theta^2+31\theta+6)-2^{16} 3^{11} x^{12}(\theta+2)(\theta+1)(2\theta+3)^2$

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 72, 1344, 48600, ...
--> OEIS
Normalized instanton numbers (n₀=1): 5, 83/2, 953/3, 5319, 97812, ... ; Common denominator:...

Discriminant

$-(4z+1)(12z+1)(3z+1)(1728z^3+864z^2+36z-1)(-1-6z-36z^2+432z^3)^2$

Local exponents

≈$-0.450956$ $-\frac{ 1}{ 3}$ $-\frac{ 1}{ 4}$ $-\frac{ 1}{ 12}$ ≈$-0.067934$ ≈$-0.061146-0.08671I$ ≈$-0.061146+0.08671I$ $0$ ≈$0.01889$ ≈$0.205625$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $1$ $1$ $1$ $1$ $0$ $1$ $1$ $\frac{ 3}{ 2}$
$1$ $1$ $1$ $1$ $1$ $3$ $3$ $0$ $1$ $3$ $\frac{ 3}{ 2}$
$2$ $2$ $2$ $2$ $2$ $4$ $4$ $0$ $2$ $4$ $2$

≈\(-0.450956\)	\(-\frac{ 1}{ 3}\)	\(-\frac{ 1}{ 4}\)	\(-\frac{ 1}{ 12}\)	≈\(-0.067934\)	≈\(-0.061146-0.08671I\)	≈\(-0.061146+0.08671I\)	\(0\)	≈\(0.01889\)	≈\(0.205625\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(\frac{ 3}{ 2}\)
\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(3\)	\(3\)	\(0\)	\(1\)	\(3\)	\(\frac{ 3}{ 2}\)
\(2\)	\(2\)	\(2\)	\(2\)	\(2\)	\(4\)	\(4\)	\(0\)	\(2\)	\(4\)	\(2\)

Note:

This is operator "12.6" from ...

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

12

New Number: 8.21 | AESZ: 251 | Superseeker: -9 -3145/3 | Hash: dd2b60d18804c72129ba319fc8b50023

Degree: 8

$\theta^4-3 x\theta(-2-11\theta-18\theta^2+27\theta^3)-2 3^{2} x^{2}\left(39\theta^4+480\theta^3+474\theta^2+276\theta+64\right)+2^{3} 3^{4} x^{3}\left(348\theta^4+1152\theta^3+1759\theta^2+1110\theta+260\right)-2^{3} 3^{5} x^{4}\left(3420\theta^4+15912\theta^3+28437\theta^2+20544\theta+5296\right)+2^{4} 3^{7} x^{5}\left(1125\theta^4+12546\theta^3+31089\theta^2+26448\theta+7480\right)+2^{5} 3^{9} x^{6}\left(1395\theta^4+3240\theta^3-3378\theta^2-7146\theta-2696\right)-2^{7} 3^{11} x^{7}\left(351\theta^4+2646\theta^3+4767\theta^2+3309\theta+800\right)-2^{7} 3^{13} x^{8}(3\theta+2)(3\theta+4)(6\theta+5)(6\theta+7)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 72, -1440, 48600, ...
--> OEIS
Normalized instanton numbers (n₀=1): -9, -27/4, -3145/3, -20907/4, -327348, ... ; Common denominator:...

Discriminant

$-(54z+1)(27z-1)(432z^2-36z+1)(-1+36z+324z^2)^2$

Local exponents

$-\frac{ 1}{ 18}-\frac{ 1}{ 18}\sqrt{ 2}$ $-\frac{ 1}{ 54}$ $0$ $-\frac{ 1}{ 18}+\frac{ 1}{ 18}\sqrt{ 2}$ $\frac{ 1}{ 27}$ $\frac{ 1}{ 24}-\frac{ 1}{ 72}\sqrt{ 3}I$ $\frac{ 1}{ 24}+\frac{ 1}{ 72}\sqrt{ 3}I$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 2}{ 3}$
$1$ $1$ $0$ $1$ $1$ $1$ $1$ $\frac{ 5}{ 6}$
$3$ $1$ $0$ $3$ $1$ $1$ $1$ $\frac{ 7}{ 6}$
$4$ $2$ $0$ $4$ $2$ $2$ $2$ $\frac{ 4}{ 3}$

\(-\frac{ 1}{ 18}-\frac{ 1}{ 18}\sqrt{ 2}\)	\(-\frac{ 1}{ 54}\)	\(0\)	\(-\frac{ 1}{ 18}+\frac{ 1}{ 18}\sqrt{ 2}\)	\(\frac{ 1}{ 27}\)	\(\frac{ 1}{ 24}-\frac{ 1}{ 72}\sqrt{ 3}I\)	\(\frac{ 1}{ 24}+\frac{ 1}{ 72}\sqrt{ 3}I\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 2}{ 3}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 5}{ 6}\)
\(3\)	\(1\)	\(0\)	\(3\)	\(1\)	\(1\)	\(1\)	\(\frac{ 7}{ 6}\)
\(4\)	\(2\)	\(0\)	\(4\)	\(2\)	\(2\)	\(2\)	\(\frac{ 4}{ 3}\)

Note:

This is operator "8.21" from ...

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

13

New Number: 8.24 | AESZ: 286 | Superseeker: 3 437/3 | Hash: 94afcd38a40c3a3e54fc3c57b4b85459

Degree: 8

$3^{2} \theta^4-3^{2} x\left(38\theta^4+82\theta^3+67\theta^2+26\theta+4\right)-3 x^{2}\left(2045\theta^4+5702\theta^3+7535\theta^2+4170\theta+852\right)+2^{3} 3 x^{3}\left(2208\theta^4+5925\theta^3+7925\theta^2+5607\theta+1512\right)+2^{3} x^{4}\left(60287\theta^4+56374\theta^3-215983\theta^2-268986\theta-85452\right)-2^{4} x^{5}\left(205651\theta^4+605608\theta^3+603579\theta^2+204622\theta+8104\right)-2^{7} x^{6}\left(51414\theta^4-273267\theta^3-502700\theta^2-305649\theta-63398\right)+2^{8} 37 x^{7}\left(7909\theta^4+18122\theta^3+17595\theta^2+8462\theta+1672\right)-2^{13} 37^{2} x^{8}(4\theta+3)(\theta+1)^2(4\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 4, 72, 1696, 49960, ...
--> OEIS
Normalized instanton numbers (n₀=1): 3, 539/24, 437/3, 18531/8, 90274/3, ... ; Common denominator:...

Discriminant

$-(-1+40z+504z^2-3088z^3+8192z^4)(-3-3z+148z^2)^2$

Local exponents

$\frac{ 3}{ 296}-\frac{ 1}{ 296}\sqrt{ 1785}$ ≈$-0.070843$ $0$ ≈$0.020383$ $\frac{ 3}{ 296}+\frac{ 1}{ 296}\sqrt{ 1785}$ ≈$0.213707$ ≈$0.213707$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 3}{ 4}$
$1$ $1$ $0$ $1$ $1$ $1$ $1$ $1$
$3$ $1$ $0$ $1$ $3$ $1$ $1$ $1$
$4$ $2$ $0$ $2$ $4$ $2$ $2$ $\frac{ 5}{ 4}$

\(\frac{ 3}{ 296}-\frac{ 1}{ 296}\sqrt{ 1785}\)	≈\(-0.070843\)	\(0\)	≈\(0.020383\)	\(\frac{ 3}{ 296}+\frac{ 1}{ 296}\sqrt{ 1785}\)	≈\(0.213707\)	≈\(0.213707\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 3}{ 4}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(3\)	\(1\)	\(0\)	\(1\)	\(3\)	\(1\)	\(1\)	\(1\)
\(4\)	\(2\)	\(0\)	\(2\)	\(4\)	\(2\)	\(2\)	\(\frac{ 5}{ 4}\)

Note:

This is operator "8.24" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-86\)	\(84\)	\(48\)	\(15\)

14

New Number: 8.54 | AESZ: | Superseeker: 0 1/3 | Hash: bb80872017d0578a4ae56172666b807c

Degree: 8

$\theta^4+x\theta(3\theta^3-6\theta^2-4\theta-1)-x^{2}\left(211\theta^4+856\theta^3+1433\theta^2+1184\theta+384\right)-2 x^{3}\left(761\theta^4+3288\theta^3+6477\theta^2+6654\theta+2700\right)+2^{2} x^{4}(\theta+1)(2013\theta^3+17379\theta^2+40726\theta+28548)+2^{3} x^{5}(\theta+1)(15719\theta^3+126105\theta^2+325408\theta+269508)+2^{5} 3^{2} x^{6}(\theta+1)(\theta+2)(1817\theta^2+11967\theta+19631)+2^{7} 3^{4} x^{7}(\theta+3)(\theta+2)(\theta+1)(89\theta+350)+2^{9} 3^{3} 43 x^{8}(\theta+1)(\theta+2)(\theta+3)(\theta+4)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 24, 72, 1296, ...
--> OEIS
Normalized instanton numbers (n₀=1): 0, 1/2, 1/3, -1, 2, ... ; Common denominator:...

Discriminant

$(4z+1)(6z+1)(43z^2+13z+1)(2z+1)^2(12z-1)^2$

Local exponents

$-\frac{ 1}{ 2}$ $-\frac{ 1}{ 4}$ $-\frac{ 1}{ 6}$ $-\frac{ 13}{ 86}-\frac{ 1}{ 86}\sqrt{ 3}I$ $-\frac{ 13}{ 86}+\frac{ 1}{ 86}\sqrt{ 3}I$ $0$ $\frac{ 1}{ 12}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $1$ $1$ $0$ $\frac{ 1}{ 2}$ $2$
$3$ $1$ $1$ $1$ $1$ $0$ $\frac{ 1}{ 2}$ $3$
$4$ $2$ $2$ $2$ $2$ $0$ $1$ $4$

\(-\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 4}\)	\(-\frac{ 1}{ 6}\)	\(-\frac{ 13}{ 86}-\frac{ 1}{ 86}\sqrt{ 3}I\)	\(-\frac{ 13}{ 86}+\frac{ 1}{ 86}\sqrt{ 3}I\)	\(0\)	\(\frac{ 1}{ 12}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(0\)	\(\frac{ 1}{ 2}\)	\(2\)
\(3\)	\(1\)	\(1\)	\(1\)	\(1\)	\(0\)	\(\frac{ 1}{ 2}\)	\(3\)
\(4\)	\(2\)	\(2\)	\(2\)	\(2\)	\(0\)	\(1\)	\(4\)

Note:

This is operator "8.54" from ...

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

15

New Number: 9.1 | AESZ: | Superseeker: -4/7 955/63 | Hash: d32eab6005ac34ecc01a9db7675daa24

Degree: 9

$7^{2} \theta^4-7 x\theta(-7-32\theta-50\theta^2+29\theta^3)+3 x^{2}\theta(532+1165\theta+512\theta^2+1235\theta^3)-2 3^{2} x^{3}\left(5373\theta^4+29040\theta^3+61493\theta^2+51786\theta+15876\right)+2^{2} 3^{3} x^{4}\left(10813\theta^4+68120\theta^3+160529\theta^2+154570\theta+53396\right)-2^{3} 3^{4} x^{5}\left(13929\theta^4+84348\theta^3+181015\theta^2+171080\theta+59172\right)+2^{5} 3^{5} x^{6}\left(6160\theta^4+35964\theta^3+69935\theta^2+58677\theta+18110\right)-2^{8} 3^{6} x^{7}\left(944\theta^4+5308\theta^3+10916\theta^2+9657\theta+3109\right)+2^{11} 3^{7} x^{8}(96\theta^2+300\theta+265)(\theta+1)^2-2^{15} 3^{9} x^{9}(\theta+1)^2(\theta+2)^2$

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 0, 72, -432, ...
--> OEIS
Normalized instanton numbers (n₀=1): -4/7, -4/7, 955/63, -262/7, -1002/7, ... ; Common denominator:...

Discriminant

$-(6z-1)(27z^2-9z+1)(192z^2+16z+1)(7-18z+144z^2)^2$

Local exponents

$-\frac{ 1}{ 24}-\frac{ 1}{ 24}\sqrt{ 2}I$ $-\frac{ 1}{ 24}+\frac{ 1}{ 24}\sqrt{ 2}I$ $0$ $\frac{ 1}{ 16}-\frac{ 1}{ 48}\sqrt{ 103}I$ $\frac{ 1}{ 16}+\frac{ 1}{ 48}\sqrt{ 103}I$ $\frac{ 1}{ 6}-\frac{ 1}{ 18}\sqrt{ 3}I$ $\frac{ 1}{ 6}$ $\frac{ 1}{ 6}+\frac{ 1}{ 18}\sqrt{ 3}I$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $0$ $1$ $1$ $1$ $1$ $1$ $1$
$1$ $1$ $0$ $3$ $3$ $1$ $1$ $1$ $2$
$2$ $2$ $0$ $4$ $4$ $2$ $2$ $2$ $2$

\(-\frac{ 1}{ 24}-\frac{ 1}{ 24}\sqrt{ 2}I\)	\(-\frac{ 1}{ 24}+\frac{ 1}{ 24}\sqrt{ 2}I\)	\(0\)	\(\frac{ 1}{ 16}-\frac{ 1}{ 48}\sqrt{ 103}I\)	\(\frac{ 1}{ 16}+\frac{ 1}{ 48}\sqrt{ 103}I\)	\(\frac{ 1}{ 6}-\frac{ 1}{ 18}\sqrt{ 3}I\)	\(\frac{ 1}{ 6}\)	\(\frac{ 1}{ 6}+\frac{ 1}{ 18}\sqrt{ 3}I\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(1\)	\(1\)	\(0\)	\(3\)	\(3\)	\(1\)	\(1\)	\(1\)	\(2\)
\(2\)	\(2\)	\(0\)	\(4\)	\(4\)	\(2\)	\(2\)	\(2\)	\(2\)

Note:

This is operator "9.1" from ...

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

16

New Number: 1.11 | AESZ: 11 | Superseeker: 324 10792428 | Hash: 8ac8b98b80383c9f0ea125ccd6e6a55d

Degree: 1

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

Maple LaTex

Coefficients of the holomorphic solution: 1, 72, 37800, 31046400, 31216185000, ...
--> OEIS
Normalized instanton numbers (n₀=1): 324, 37260, 10792428, 4580482284, 2405245303584, ... ; Common denominator:...

Discriminant

$1-1728z$

Local exponents

$0$ $\frac{ 1}{ 1728}$ $\infty$
$0$ $0$ $\frac{ 1}{ 4}$
$0$ $1$ $\frac{ 1}{ 3}$
$0$ $1$ $\frac{ 2}{ 3}$
$0$ $2$ $\frac{ 3}{ 4}$

\(0\)	\(\frac{ 1}{ 1728}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 1}{ 4}\)
\(0\)	\(1\)	\(\frac{ 1}{ 3}\)
\(0\)	\(1\)	\(\frac{ 2}{ 3}\)
\(0\)	\(2\)	\(\frac{ 3}{ 4}\)

Note:

A-incarnation: X(4,6) in P^5(1,1,1,2,2,3)

\(-\frac{ 1}{ 2}+24\lambda\)	\(12\lambda\)	\(-\frac{ 1}{ 8}+2\lambda\)	\(-.1127229e-2\)
\(-1\)	\(\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 12}\)	\(-\frac{ 1}{ 8}-2\lambda\)
\(-6\)	\(0\)	\(\frac{ 1}{ 2}\)	\(-12\lambda\)
\(-12\)	\(-6\)	\(-1\)	\(-\frac{ 1}{ 2}-24\lambda\)

\(-1.363320470+.333408287I\)	\(-2.378556075+\frac{ 1}{ 4}I2^{ \frac{ 1}{ 4}}Pi^{ \frac{ 3}{ 2}}ln(2)^(\frac{ 7}{ 4})\)	\(1.952097134+1.450430332I\)	\(.352541506-.730455452I\)
\(-2.231842169+5.463251200I\)	\(4.752377305+8.996899861I\)	\(5.529765042-2.362775601I\)	\(-1.121053097-1.662936757I\)
\(-1.857729799+3.694349576I\)	\(2.129374012+I7^{ \frac{ 7}{ 8}}ln(2)^{ \frac{ 1}{ 2}}ln(3)^(\frac{ 1}{ 6})\)	\(4.991998028-1.348211717I\)	\(-.692201690-1.222934527I\)
\(-10.712842412+26.223605760I\)	\(18.011411066+43.185119335I\)	\(26.542872203-11.341322886I\)	\(-4.381054863-7.982096432I\)

\(\frac{ 3}{ 2}-24\lambda\)	\(0\)	\(\frac{ 1}{ 2}\)	\(1\)
\(1\)	\(0\)	\(-1\)	\(0\)
\(6\)	\(-12\)	\(0\)	\(0\)
\(12\)	\(0\)	\(0\)	\(0\)

\(1+344\lambda\)	\(0\)	\(\frac{ 215}{ 3}\lambda\)	\(.115791449\)
\(5\)	\(1\)	\(\frac{ 25}{ 24}\)	\(-\frac{ 215}{ 3}\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(24\)	\(0\)	\(5\)	\(1-344\lambda\)

\(-344\lambda\)	\(9\)	\(1\)	\(1\)
\(-5\)	\(-12\)	\(-1\)	\(0\)
\(0\)	\(24\)	\(0\)	\(0\)
\(-24\)	\(0\)	\(0\)	\(0\)

Calabi-Yau differential operator database v.3

Summary

Discriminant

Local exponents

Explicit solution

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:

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

Discriminant

Local exponents

Note:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Explicit solution

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

Discriminant

Local exponents

Note:

Characteristic classes:

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

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

Discriminant

Local exponents

Note:

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:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

Discriminant

Local exponents

Note:

Monodromy (with respect to Frobenius basis)