Calabi-Yau differential operator database v.3.0

1

New Number: 2.28 | AESZ: 49 | Superseeker: 48 2864 | Hash: 0a357a8c4fd703ab062148eadcd94daa

Degree: 2

$\theta^4-2^{2} 3 x(2\theta+1)^2(18\theta^2+18\theta+7)+2^{4} 3^{6} x^{2}(2\theta+1)^2(2\theta+3)^2$

Coefficients of the holomorphic solution: 1, 84, 17820, 4868400, 1499003100, ...
--> OEIS
Normalized instanton numbers (n₀=1): 48, -438, 2864, 77958, -4942032, ... ; Common denominator:...

Discriminant

$(432z-1)^2$

Local exponents

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

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

Note:

Hadamard product $A \ast$

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

2

New Number: 5.2 | AESZ: 19 | Superseeker: 80/23 4655/23 | Hash: 4532f44d62f644bf66aa7b153d4f5c5a

Degree: 5

$23^{2} \theta^4-23 x\left(921\theta^4+2046\theta^3+1644\theta^2+621\theta+92\right)-x^{2}\left(380851\theta^4+1328584\theta^3+1772673\theta^2+1033528\theta+221168\right)-2 x^{3}\left(475861\theta^4+1310172\theta^3+1028791\theta^2+208932\theta-27232\right)-2^{2} 17 x^{4}\left(8873\theta^4+14020\theta^3+5139\theta^2-1664\theta-976\right)+2^{3} 3 17^{2} x^{5}(\theta+1)^2(3\theta+2)(3\theta+4)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 4, 84, 2200, 71140, ...
--> OEIS
Normalized instanton numbers (n₀=1): 80/23, 1157/46, 4655/23, 71184/23, 1156690/23, ... ; Common denominator:...

Discriminant

$(54z-1)(z^2-11z-1)(23+34z)^2$

Local exponents

$-\frac{ 23}{ 34}$ $\frac{ 11}{ 2}-\frac{ 5}{ 2}\sqrt{ 5}$ $0$ $\frac{ 1}{ 54}$ $\frac{ 11}{ 2}+\frac{ 5}{ 2}\sqrt{ 5}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 2}{ 3}$
$1$ $1$ $0$ $1$ $1$ $1$
$3$ $1$ $0$ $1$ $1$ $1$
$4$ $2$ $0$ $2$ $2$ $\frac{ 4}{ 3}$

\(-\frac{ 23}{ 34}\)	\(\frac{ 11}{ 2}-\frac{ 5}{ 2}\sqrt{ 5}\)	\(0\)	\(\frac{ 1}{ 54}\)	\(\frac{ 11}{ 2}+\frac{ 5}{ 2}\sqrt{ 5}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 2}{ 3}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(3\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(4\)	\(2\)	\(0\)	\(2\)	\(2\)	\(\frac{ 4}{ 3}\)

Note:

This is operator "5.2" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-106\)	\(88\)	\(46\)	\(15\)

3

New Number: 5.80 | AESZ: 311 | Superseeker: 25/13 875/13 | Hash: 8219f3f4bd56f6c2b2cc3ab9093b65d1

Degree: 5

$13^{2} \theta^4-13 x\left(327\theta^4+1038\theta^3+857\theta^2+338\theta+52\right)-2^{4} x^{2}\left(12848\theta^4+42008\theta^3+52082\theta^2+28548\theta+5707\right)-2^{11} x^{3}\left(122\theta^4-1872\theta^3-6341\theta^2-5772\theta-1547\right)+2^{16} x^{4}(2\theta+1)(76\theta^3+426\theta^2+570\theta+227)+2^{23} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 4, 84, 1840, 56980, ...
--> OEIS
Normalized instanton numbers (n₀=1): 25/13, 1359/52, 875/13, 36572/13, 256800/13, ... ; Common denominator:...

Discriminant

$(8192z^3-896z^2-35z+1)(13+64z)^2$

Local exponents

$-\frac{ 13}{ 64}$ ≈$-0.045147$ $0$ ≈$0.020117$ ≈$0.134405$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $1$ $0$ $1$ $1$ $1$
$3$ $1$ $0$ $1$ $1$ $1$
$4$ $2$ $0$ $2$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 13}{ 64}\)	≈\(-0.045147\)	\(0\)	≈\(0.020117\)	≈\(0.134405\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(3\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(4\)	\(2\)	\(0\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.80" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-144\)	\(100\)	\(52\)	\(17\)

4

New Number: 14.2 | AESZ: | Superseeker: 27/5 1619/5 | Hash: c0f6d85270164c8c5a63d1bb2deaba83

Degree: 14

\(5^{2} \theta^4+3^{2} 5 x\theta(6\theta^3-36\theta^2-23\theta-5)-x^{2}\left(43856\theta^4+189068\theta^3+226691\theta^2+135510\theta+33600\right)-3^{2} x^{3}\left(224236\theta^4+916896\theta^3+1403247\theta^2+1048995\theta+313920\right)-x^{4}\left(44621090\theta^4+199900036\theta^3+357072757\theta^2+304636250\theta+101358144\right)-3^{2} x^{5}\left(69593744\theta^4+347076728\theta^3+696076003\theta^2+653370139\theta+234075456\right)-3^{2} x^{6}\left(681084088\theta^4+3766244020\theta^3+8299124637\theta^2+8400442322\theta+3184811840\right)-3^{3} x^{7}\left(1616263276\theta^4+9835107968\theta^3+23484467027\theta^2+25311872719\theta+10046134656\right)-3^{3} x^{8}\left(8527956293\theta^4+56671723156\theta^3+145225420081\theta^2+165230257706\theta+68152357440\right)-2 3^{4} x^{9}\left(5575274615\theta^4+40185448970\theta^3+109721715457\theta^2+130944512374\theta+55834822464\right)-2^{3} 3^{3} x^{10}\left(12062719219\theta^4+93737716664\theta^3+271167874625\theta^2+337796659588\theta+148305175248\right)-2^{5} 3^{5} x^{11}(\theta+1)(691573543\theta^3+5071601663\theta^2+12510902832\theta+10260936720)-2^{7} 3^{6} x^{12}(\theta+1)(\theta+2)(80620421\theta^2+475174733\theta+711172676)-2^{14} 3^{6} 5 x^{13}(\theta+3)(\theta+2)(\theta+1)(107069\theta+369433)-2^{19} 3^{8} 5^{2} 29 x^{14}(\theta+1)(\theta+2)(\theta+3)(\theta+4)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 84, 1944, 70476, ...
--> OEIS
Normalized instanton numbers (n₀=1): 27/5, 158/5, 1619/5, 51193/10, 485082/5, ... ; Common denominator:...

Discriminant

$-(9z+1)(6z+1)(348z^2+51z-1)(5z+1)^2(4z+1)^2(576z^3+357z^2+72z+5)^2$

Local exponents

≈$-0.298314$ $-\frac{ 1}{ 4}$ $-\frac{ 1}{ 5}$ $-\frac{ 1}{ 6}$ $-\frac{ 17}{ 232}-\frac{ 11}{ 696}\sqrt{ 33}$ ≈$-0.160739-0.057112I$ ≈$-0.160739+0.057112I$ $-\frac{ 1}{ 9}$ $0$ $-\frac{ 17}{ 232}+\frac{ 11}{ 696}\sqrt{ 33}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $\frac{ 1}{ 2}$ $\frac{ 1}{ 2}$ $1$ $1$ $1$ $1$ $1$ $0$ $1$ $2$
$3$ $\frac{ 1}{ 2}$ $\frac{ 1}{ 2}$ $1$ $1$ $3$ $3$ $1$ $0$ $1$ $3$
$4$ $1$ $1$ $2$ $2$ $4$ $4$ $2$ $0$ $2$ $4$

≈\(-0.298314\)	\(-\frac{ 1}{ 4}\)	\(-\frac{ 1}{ 5}\)	\(-\frac{ 1}{ 6}\)	\(-\frac{ 17}{ 232}-\frac{ 11}{ 696}\sqrt{ 33}\)	≈\(-0.160739-0.057112I\)	≈\(-0.160739+0.057112I\)	\(-\frac{ 1}{ 9}\)	\(0\)	\(-\frac{ 17}{ 232}+\frac{ 11}{ 696}\sqrt{ 33}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(\frac{ 1}{ 2}\)	\(\frac{ 1}{ 2}\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(2\)
\(3\)	\(\frac{ 1}{ 2}\)	\(\frac{ 1}{ 2}\)	\(1\)	\(1\)	\(3\)	\(3\)	\(1\)	\(0\)	\(1\)	\(3\)
\(4\)	\(1\)	\(1\)	\(2\)	\(2\)	\(4\)	\(4\)	\(2\)	\(0\)	\(2\)	\(4\)

Note:

This is operator "14.2" from ...

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

5

New Number: 6.35 | AESZ: | Superseeker: | Hash: de26083962cade55a4938b4011d0008e

Degree: 6

$\theta^4-3 x\left(63\theta^4+234\theta^3+247\theta^2+130\theta+28\right)+2 3^{4} x^{2}\left(9\theta^4+522\theta^3+1207\theta^2+1058\theta+356\right)+2^{2} 3^{7} x^{3}\left(135\theta^4+270\theta^3-730\theta^2-1395\theta-696\right)-2^{3} 3^{10} x^{4}\left(63\theta^4+774\theta^3+1372\theta^2+817\theta+88\right)-2^{4} 3^{13} x^{5}\left(72\theta^4+72\theta^3-325\theta^2-629\theta-308\right)+2^{5} 3^{16} x^{6}(3\theta+5)(3\theta+4)(2\theta+3)^2$

Maple LaTex

Coefficients of the holomorphic solution: 1, 84, 7452, 692688, 66448116, ...
--> OEIS
Normalized instanton numbers (n₀=1): 21, -1617/4, 7941, -986355/4, 8179455, ... ; Common denominator:...

Discriminant

$(54z-1)(27z-1)(54z+1)^2(108z-1)^2$

Local exponents

$-\frac{ 1}{ 54}$ $0$ $\frac{ 1}{ 108}$ $\frac{ 1}{ 54}$ $\frac{ 1}{ 27}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 4}{ 3}$
$1$ $0$ $-\frac{ 1}{ 2}$ $1$ $1$ $\frac{ 3}{ 2}$
$3$ $0$ $1$ $1$ $1$ $\frac{ 3}{ 2}$
$4$ $0$ $\frac{ 3}{ 2}$ $2$ $2$ $\frac{ 5}{ 3}$

\(-\frac{ 1}{ 54}\)	\(0\)	\(\frac{ 1}{ 108}\)	\(\frac{ 1}{ 54}\)	\(\frac{ 1}{ 27}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 4}{ 3}\)
\(1\)	\(0\)	\(-\frac{ 1}{ 2}\)	\(1\)	\(1\)	\(\frac{ 3}{ 2}\)
\(3\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 3}{ 2}\)
\(4\)	\(0\)	\(\frac{ 3}{ 2}\)	\(2\)	\(2\)	\(\frac{ 5}{ 3}\)

Note:

This is operator "6.35" from ...

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

6

New Number: 24.16 | AESZ: | Superseeker: 20/3 7540/81 | Hash: 9b288a17ff21e31512fe3730227152b2

Degree: 24

\(3^{3} \theta^4-2^{2} 3^{2} x\left(51\theta^4+94\theta^3+81\theta^2+34\theta+6\right)+2^{4} 3 x^{2}\left(777\theta^4+3952\theta^3+5495\theta^2+3326\theta+840\right)+2^{6} x^{3}\left(6076\theta^4-36580\theta^3-96208\theta^2-73820\theta-22971\right)-2^{8} x^{4}\left(91962\theta^4+92760\theta^3-287340\theta^2-275194\theta-88617\right)+2^{10} x^{5}\left(255290\theta^4+840624\theta^3-841718\theta^2-1139664\theta-435957\right)+2^{12} x^{6}\left(344350\theta^4-2037616\theta^3+2302248\theta^2+3899366\theta+1855473\right)-2^{14} x^{7}\left(3596934\theta^4-1919904\theta^3+4846554\theta^2+7817604\theta+3815689\right)+2^{16} x^{8}\left(8122336\theta^4-2455824\theta^3+1743468\theta^2+4576350\theta+2211269\right)-2^{18} x^{9}\left(4736512\theta^4-15179116\theta^3-30014900\theta^2-27517140\theta-10151379\right)-2^{20} x^{10}\left(10049328\theta^4+43157856\theta^3+84407914\theta^2+77408798\theta+27100963\right)+2^{22} x^{11}\left(16849298\theta^4+43081512\theta^3+66129374\theta^2+42478644\theta+5863649\right)-2^{24} x^{12}\left(2120418\theta^4-19095648\theta^3-77625292\theta^2-120132994\theta-66210663\right)-2^{26} x^{13}\left(11329982\theta^4+59772152\theta^3+148599750\theta^2+178534328\theta+82729387\right)+2^{28} x^{14}\left(5764462\theta^4+18211776\theta^3+27275956\theta^2+15555738\theta+1058615\right)+2^{30} x^{15}\left(2768822\theta^4+19152112\theta^3+48724938\theta^2+52160308\theta+17451325\right)-2^{32} x^{16}\left(2326731\theta^4+9250224\theta^3+12025644\theta^2-5312634\theta-13944959\right)-2^{34} x^{17}\left(238805\theta^4+1852302\theta^3+1102491\theta^2-7080654\theta-8783903\right)+2^{36} x^{18}\left(377045\theta^4+632272\theta^3-4486531\theta^2-16948308\theta-14739201\right)+2^{39} x^{19}\left(16533\theta^4+234894\theta^3+875221\theta^2+1653596\theta+1206823\right)-2^{42} x^{20}\left(5449\theta^4-48270\theta^3-337532\theta^2-586347\theta-302891\right)-2^{44} x^{21}\left(2935\theta^4+30482\theta^3+120769\theta^2+177402\theta+84039\right)-2^{46} x^{22}\left(179\theta^4+6980\theta^3+3029\theta^2+47490\theta+25879\right)+2^{49} 5 x^{23}\left(47\theta^4+378\theta^3+1119\theta^2+1464\theta+719\right)+2^{52} 5^{2} x^{24}\left((\theta+2)^4\right)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 84, 864, 4164, ...
--> OEIS
Normalized instanton numbers (n₀=1): 20/3, -194/9, 7540/81, -2021323/972, 1669749088/91125, ... ; Common denominator:...

Discriminant

$(4z+1)(16z^2+4z-1)(16384z^6+10240z^5+9984z^4+1024z^3-1056z^2+56z-1)(4z-1)^2(10240z^5-8704z^4+2752z^3-176z^2-4z+1)^2(4z+3)^3$

No data for singularities

Note:

This is operator "24.16" from ...

Calabi-Yau differential operator database v.3

Summary

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:

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

Note:

Monodromy (with respect to Frobenius basis)

Basis of the Doran-Morgan lattice

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

\(-\frac{ 1}{ 2}+168\lambda\)	\(\frac{ 1}{ 4}+84\lambda\)	\(0\)	\(.7265788e-2\)
\(0\)	\(1\)	\(0\)	\(0\)
\(-6\)	\(1\)	\(1\)	\(\frac{ 1}{ 4}-84\lambda\)
\(-12\)	\(-6\)	\(0\)	\(-\frac{ 1}{ 2}-168\lambda\)

\(\frac{ 3}{ 2}-168\lambda\)	\(1\)	\(\frac{ 1}{ 2}\)	\(1\)
\(0\)	\(0\)	\(-1\)	\(0\)
\(6\)	\(-12\)	\(0\)	\(0\)
\(12\)	\(0\)	\(0\)	\(0\)