Calabi-Yau differential operator database v.3.0

1

New Number: 2.3 | AESZ: 68 | Superseeker: 52 220220 | Hash: 13a48045ff0a42a9fcfbdb710baf1997

Degree: 2

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

Coefficients of the holomorphic solution: 1, 24, 4200, 1034880, 311711400, ...
--> OEIS
Normalized instanton numbers (n₀=1): 52, 2814, 220220, 29135058, 4512922272, ... ; Common denominator:...

Discriminant

\(-(64z+1)(512z-1)\)

Local exponents

\(-\frac{ 1}{ 64}\) \(0\) \(\frac{ 1}{ 512}\) \(\infty\)
\(0\) \(0\) \(0\) \(\frac{ 1}{ 4}\)
\(1\) \(0\) \(1\) \(\frac{ 3}{ 4}\)
\(1\) \(0\) \(1\) \(\frac{ 5}{ 4}\)
\(2\) \(0\) \(2\) \(\frac{ 7}{ 4}\)

\(-\frac{ 1}{ 64}\)	\(0\)	\(\frac{ 1}{ 512}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 4}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 3}{ 4}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 5}{ 4}\)
\(2\)	\(0\)	\(2\)	\(\frac{ 7}{ 4}\)

Note:

C*a

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-228\)	\(72\)	\(12\)	\(8\)

2

New Number: 3.1 | AESZ: 34 | Superseeker: 1 28/3 | Hash: e5461c5f5ae4d929328f66b8955a31f5

Degree: 3

\(\theta^4-x\left(35\theta^4+70\theta^3+63\theta^2+28\theta+5\right)+x^{2}(\theta+1)^2(259\theta^2+518\theta+285)-3^{2} 5^{2} x^{3}(\theta+1)^2(\theta+2)^2\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 5, 45, 545, 7885, ...
--> OEIS
Normalized instanton numbers (n₀=1): 1, 2, 28/3, 52, 350, ... ; Common denominator:...

Discriminant

\(-(z-1)(25z-1)(9z-1)\)

Local exponents

\(0\) \(\frac{ 1}{ 25}\) \(\frac{ 1}{ 9}\) \(1\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(1\)
\(0\) \(1\) \(1\) \(1\) \(1\)
\(0\) \(1\) \(1\) \(1\) \(2\)
\(0\) \(2\) \(2\) \(2\) \(2\)

\(0\)	\(\frac{ 1}{ 25}\)	\(\frac{ 1}{ 9}\)	\(1\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)	\(2\)
\(0\)	\(2\)	\(2\)	\(2\)	\(2\)

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-80\)	\(120\)	\(120\)	\(30\)

3

New Number: 12.17 | AESZ: | Superseeker: 4 52 | Hash: e65be092d4832d3740d2a3078755f447

Degree: 12

\(\theta^4+2^{2} x\left(24\theta^4+6\theta^3+11\theta^2+8\theta+2\right)+2^{4} x^{2}\left(209\theta^4+2\theta^3+23\theta^2-10\right)+2^{7} x^{3}\left(223\theta^4-1218\theta^3-2225\theta^2-2088\theta-776\right)-2^{10} x^{4}\left(1409\theta^4+9634\theta^3+19337\theta^2+18420\theta+6872\right)-2^{13} x^{5}\left(6527\theta^4+35858\theta^3+78357\theta^2+78428\theta+30414\right)-2^{17} x^{6}\left(6276\theta^4+37704\theta^3+91143\theta^2+97914\theta+40036\right)-2^{21} x^{7}\left(2923\theta^4+22130\theta^3+61939\theta^2+73401\theta+32138\right)-2^{24} x^{8}\left(602\theta^4+10928\theta^3+42765\theta^2+60182\theta+29287\right)+2^{26} x^{9}\left(2352\theta^4+7392\theta^3-7024\theta^2-31968\theta-21891\right)+2^{29} x^{10}\left(1584\theta^4+11904\theta^3+24696\theta^2+19776\theta+4915\right)-2^{35} x^{11}\left(16\theta^4-176\theta^3-784\theta^2-1036\theta-449\right)-2^{39} x^{12}\left((2\theta+3)^4\right)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, -8, 112, -1152, 19216, ...
--> OEIS
Normalized instanton numbers (n₀=1): 4, 7/2, 52, 500, 2796, ... ; Common denominator:...

Discriminant

\(-(8z+1)(256z^2+16z-1)(1024z^3-160z^2-28z-1)^2(16z+1)^3\)

Local exponents

\(-\frac{ 1}{ 8}\) \(-\frac{ 1}{ 32}-\frac{ 1}{ 32}\sqrt{ 5}\) \(-\frac{ 1}{ 16}\) ≈\(-0.057187-0.018391I\) ≈\(-0.057187+0.018391I\) \(0\) \(-\frac{ 1}{ 32}+\frac{ 1}{ 32}\sqrt{ 5}\) ≈\(0.270624\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(\frac{ 3}{ 2}\)
\(1\) \(1\) \(0\) \(1\) \(1\) \(0\) \(1\) \(1\) \(\frac{ 3}{ 2}\)
\(1\) \(1\) \(0\) \(3\) \(3\) \(0\) \(1\) \(3\) \(\frac{ 3}{ 2}\)
\(2\) \(2\) \(0\) \(4\) \(4\) \(0\) \(2\) \(4\) \(\frac{ 3}{ 2}\)

\(-\frac{ 1}{ 8}\)	\(-\frac{ 1}{ 32}-\frac{ 1}{ 32}\sqrt{ 5}\)	\(-\frac{ 1}{ 16}\)	≈\(-0.057187-0.018391I\)	≈\(-0.057187+0.018391I\)	\(0\)	\(-\frac{ 1}{ 32}+\frac{ 1}{ 32}\sqrt{ 5}\)	≈\(0.270624\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 3}{ 2}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(\frac{ 3}{ 2}\)
\(1\)	\(1\)	\(0\)	\(3\)	\(3\)	\(0\)	\(1\)	\(3\)	\(\frac{ 3}{ 2}\)
\(2\)	\(2\)	\(0\)	\(4\)	\(4\)	\(0\)	\(2\)	\(4\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "12.17" from ...

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

4

New Number: 13.3 | AESZ: | Superseeker: 4 52 | Hash: 9127ce057848ca38f220a7bb67e245a2

Degree: 13

\(\theta^4-2^{2} x\left(38\theta^4+50\theta^3+53\theta^2+28\theta+6\right)+2^{4} x^{2}\left(617\theta^4+1598\theta^3+2361\theta^2+1812\theta+586\right)-2^{8} x^{3}\left(1422\theta^4+5468\theta^3+10321\theta^2+9918\theta+3961\right)+2^{11} x^{4}\left(4165\theta^4+21060\theta^3+48228\theta^2+54855\theta+25440\right)-2^{14} x^{5}\left(8248\theta^4+50660\theta^3+135119\theta^2+175776\theta+91644\right)+2^{16} x^{6}\left(23161\theta^4+161282\theta^3+479205\theta^2+690060\theta+393943\right)-2^{20} x^{7}\left(12116\theta^4+89614\theta^3+279997\theta^2+425868\theta+256804\right)+2^{23} x^{8}\left(9924\theta^4+74644\theta^3+231233\theta^2+346097\theta+206261\right)-2^{27} x^{9}\left(3250\theta^4+24820\theta^3+75837\theta^2+107033\theta+58293\right)+2^{28} x^{10}\left(6672\theta^4+52000\theta^3+164304\theta^2+235440\theta+126113\right)-2^{32} x^{11}\left(1312\theta^4+10208\theta^3+32688\theta^2+49072\theta+28407\right)+2^{36} x^{12}\left(192\theta^4+1568\theta^3+4952\theta^2+7144\theta+3959\right)-2^{40} x^{13}\left((2\theta+5)^4\right)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 24, 464, 8832, 178960, ...
--> OEIS
Normalized instanton numbers (n₀=1): 4, 7/2, 52, 500, 2796, ... ; Common denominator:...

Discriminant

\(-(1-48z+256z^2)(8z-1)^2(512z^3-32z^2+20z-1)^2(16z-1)^3\)

Local exponents

\(0\) ≈\(0.005863-0.196043I\) ≈\(0.005863+0.196043I\) \(\frac{ 3}{ 32}-\frac{ 1}{ 32}\sqrt{ 5}\) ≈\(0.050774\) \(\frac{ 1}{ 16}\) \(\frac{ 1}{ 8}\) \(\frac{ 3}{ 32}+\frac{ 1}{ 32}\sqrt{ 5}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(\frac{ 5}{ 2}\)
\(0\) \(1\) \(1\) \(1\) \(1\) \(0\) \(0\) \(1\) \(\frac{ 5}{ 2}\)
\(0\) \(3\) \(3\) \(1\) \(3\) \(0\) \(-1\) \(1\) \(\frac{ 5}{ 2}\)
\(0\) \(4\) \(4\) \(2\) \(4\) \(0\) \(1\) \(2\) \(\frac{ 5}{ 2}\)

\(0\)	≈\(0.005863-0.196043I\)	≈\(0.005863+0.196043I\)	\(\frac{ 3}{ 32}-\frac{ 1}{ 32}\sqrt{ 5}\)	≈\(0.050774\)	\(\frac{ 1}{ 16}\)	\(\frac{ 1}{ 8}\)	\(\frac{ 3}{ 32}+\frac{ 1}{ 32}\sqrt{ 5}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 5}{ 2}\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(0\)	\(0\)	\(1\)	\(\frac{ 5}{ 2}\)
\(0\)	\(3\)	\(3\)	\(1\)	\(3\)	\(0\)	\(-1\)	\(1\)	\(\frac{ 5}{ 2}\)
\(0\)	\(4\)	\(4\)	\(2\)	\(4\)	\(0\)	\(1\)	\(2\)	\(\frac{ 5}{ 2}\)

Note:

This is operator "13.3" from ...

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

5

New Number: 8.69 | AESZ: | Superseeker: 4 52 | Hash: e303d10e77a367612be2fb706f37b895

Degree: 8

\(\theta^4-2^{2} x\left(20\theta^4+34\theta^3+29\theta^2+12\theta+2\right)+2^{4} x^{2}\left(125\theta^4+362\theta^3+471\theta^2+284\theta+66\right)-2^{7} x^{3}\left(191\theta^4+606\theta^3+855\theta^2+588\theta+154\right)+2^{10} x^{4}\left(192\theta^4+552\theta^3+562\theta^2+268\theta+49\right)-2^{13} x^{5}\left(134\theta^4+380\theta^3+373\theta^2+124\theta+3\right)+2^{16} x^{6}\left(61\theta^4+150\theta^3+173\theta^2+93\theta+19\right)-2^{19} x^{7}\left(19\theta^4+50\theta^3+56\theta^2+31\theta+7\right)+2^{23} x^{8}\left((\theta+1)^4\right)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 8, 128, 2816, 74896, ...
--> OEIS
Normalized instanton numbers (n₀=1): 4, 15/2, 52, 1563/2, 7276, ... ; Common denominator:...

Discriminant

\((16z-1)(8z-1)(64z^2-48z+1)(1-4z+32z^2)^2\)

Local exponents

\(0\) \(\frac{ 3}{ 8}-\frac{ 1}{ 4}\sqrt{ 2}\) \(\frac{ 1}{ 16}-\frac{ 1}{ 16}\sqrt{ 7}I\) \(\frac{ 1}{ 16}\) \(\frac{ 1}{ 16}+\frac{ 1}{ 16}\sqrt{ 7}I\) \(\frac{ 1}{ 8}\) \(\frac{ 3}{ 8}+\frac{ 1}{ 4}\sqrt{ 2}\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(1\)
\(0\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\) \(1\)
\(0\) \(1\) \(3\) \(1\) \(3\) \(1\) \(1\) \(1\)
\(0\) \(2\) \(4\) \(2\) \(4\) \(2\) \(2\) \(1\)

\(0\)	\(\frac{ 3}{ 8}-\frac{ 1}{ 4}\sqrt{ 2}\)	\(\frac{ 1}{ 16}-\frac{ 1}{ 16}\sqrt{ 7}I\)	\(\frac{ 1}{ 16}\)	\(\frac{ 1}{ 16}+\frac{ 1}{ 16}\sqrt{ 7}I\)	\(\frac{ 1}{ 8}\)	\(\frac{ 3}{ 8}+\frac{ 1}{ 4}\sqrt{ 2}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(3\)	\(1\)	\(3\)	\(1\)	\(1\)	\(1\)
\(0\)	\(2\)	\(4\)	\(2\)	\(4\)	\(2\)	\(2\)	\(1\)

Note:

This is operator "8.69" from ...

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

6

New Number: 8.78 | AESZ: | Superseeker: 52 48732 | Hash: 2fb524ad6efb19e0117ae7acbd9f67b9

Degree: 8

\(\theta^4-2^{2} x\left(184\theta^4+224\theta^3+175\theta^2+63\theta+9\right)+2^{4} 3 x^{2}\left(3472\theta^4+9664\theta^3+9864\theta^2+4264\theta+705\right)-2^{8} 3^{2} x^{3}\left(1936\theta^4+27936\theta^3+43336\theta^2+21528\theta+3933\right)-2^{16} 3^{3} x^{4}\left(1384\theta^4+524\theta^3-4555\theta^2-3404\theta-753\right)+2^{19} 3^{4} x^{5}\left(3440\theta^4+13712\theta^3-58\theta^2-3774\theta-1161\right)+2^{22} 3^{5} x^{6}\left(11312\theta^4-9888\theta^3-10808\theta^2-1608\theta+459\right)-2^{26} 3^{7} x^{7}(2\theta+1)(1336\theta^3+2772\theta^2+2234\theta+663)-2^{32} 3^{9} x^{8}(2\theta+1)(4\theta+3)(4\theta+5)(2\theta+3)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 36, 3780, 555120, 95199300, ...
--> OEIS
Normalized instanton numbers (n₀=1): 52, -399, 48732, -992750, 98106208, ... ; Common denominator:...

Discriminant

\(-(256z-1)(110592z^3+6912z^2-288z+1)(-1+96z+13824z^2)^2\)

Local exponents

≈\(-0.091906\) \(-\frac{ 1}{ 288}-\frac{ 1}{ 288}\sqrt{ 7}\) \(0\) ≈\(0.00385\) \(\frac{ 1}{ 256}\) \(-\frac{ 1}{ 288}+\frac{ 1}{ 288}\sqrt{ 7}\) ≈\(0.025556\) \(\infty\)
\(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(0\) \(\frac{ 1}{ 2}\)
\(1\) \(1\) \(0\) \(1\) \(1\) \(1\) \(1\) \(\frac{ 3}{ 4}\)
\(1\) \(3\) \(0\) \(1\) \(1\) \(3\) \(1\) \(\frac{ 5}{ 4}\)
\(2\) \(4\) \(0\) \(2\) \(2\) \(4\) \(2\) \(\frac{ 3}{ 2}\)

≈\(-0.091906\)	\(-\frac{ 1}{ 288}-\frac{ 1}{ 288}\sqrt{ 7}\)	\(0\)	≈\(0.00385\)	\(\frac{ 1}{ 256}\)	\(-\frac{ 1}{ 288}+\frac{ 1}{ 288}\sqrt{ 7}\)	≈\(0.025556\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 3}{ 4}\)
\(1\)	\(3\)	\(0\)	\(1\)	\(1\)	\(3\)	\(1\)	\(\frac{ 5}{ 4}\)
\(2\)	\(4\)	\(0\)	\(2\)	\(2\)	\(4\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "8.78" from ...

\(4.+2.209786237I\)	\(-1.5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000-1.104893119I\)	\(1+152\lambda\)	\(-.171535199-114\lambda\)
\(8\)	\(-3\)	\(\frac{ 8}{ 3}\)	\(-1-152\lambda\)
\(12\)	\(-6\)	\(5\)	\(-1.5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000-1.104893119I\)
\(24\)	\(-12\)	\(8\)	\(-2.-2.209786237I\)

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

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

\(1.+1.104893119I\)	\(0\)	\(57\lambda\)	\(\frac{ 276}{ 2713}\)
\(3\)	\(1\)	\(\frac{ 3}{ 4}\)	\(-57\lambda\)
\(0\)	\(0\)	\(1\)	\(0\)
\(12\)	\(0\)	\(3\)	\(1.-1.104893119I\)

\(-\frac{ 1104893119}{ 1000000000}I\)	\(5\)	\(1\)	\(1\)
\(-3\)	\(-6\)	\(-1\)	\(0\)
\(0\)	\(12\)	\(0\)	\(0\)
\(-12\)	\(0\)	\(0\)	\(0\)

\(-80\lambda\)	\(25\)	\(1\)	\(1\)
\(-5\)	\(-60\)	\(-1\)	\(0\)
\(0\)	\(120\)	\(0\)	\(0\)
\(-120\)	\(0\)	\(0\)	\(0\)