Calabi-Yau differential operator database v.3.0

1

New Number: 4.34 | AESZ: 99 | Superseeker: 647/13 942613/13 | Hash: f6c6b846edc829f336d8e4ae1dcb5618

Degree: 4

$13^{2} \theta^4-13 x\left(4569\theta^4+9042\theta^3+6679\theta^2+2158\theta+260\right)+2^{4} x^{2}\left(6386\theta^4-1774\theta^3-17898\theta^2-11596\theta-2119\right)+2^{8} x^{3}\left(67\theta^4+1248\theta^3+1091\theta^2+312\theta+26\right)-2^{12} x^{4}\left((2\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 20, 2196, 369200, 75562900, ...
--> OEIS
Normalized instanton numbers (n₀=1): 647/13, 16166/13, 942613/13, 80218296/13, 8418215008/13, ... ; Common denominator:...

Discriminant

$-(256z^2+349z-1)(-13+16z)^2$

Local exponents

$-\frac{ 349}{ 512}-\frac{ 85}{ 512}\sqrt{ 17}$ $0$ $s_1$ $s_2$ $-\frac{ 349}{ 512}+\frac{ 85}{ 512}\sqrt{ 17}$ $\frac{ 13}{ 16}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$ $1$ $1$ $\frac{ 1}{ 2}$
$1$ $0$ $1$ $1$ $1$ $3$ $\frac{ 1}{ 2}$
$2$ $0$ $2$ $2$ $2$ $4$ $\frac{ 1}{ 2}$

\(-\frac{ 349}{ 512}-\frac{ 85}{ 512}\sqrt{ 17}\)	\(0\)	\(s_1\)	\(s_2\)	\(-\frac{ 349}{ 512}+\frac{ 85}{ 512}\sqrt{ 17}\)	\(\frac{ 13}{ 16}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(3\)	\(\frac{ 1}{ 2}\)
\(2\)	\(0\)	\(2\)	\(2\)	\(2\)	\(4\)	\(\frac{ 1}{ 2}\)

Note:

Sporadic Operator.
There is a second MUM point hidden at infinity. That is operator AESZ 207/4.38
A-Incarnation: $5 \times 5$-Pfaffian in P^5

A-Incarnation: 5 \times 5 Pfaffian in P^5

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-120\)	\(58\)	\(13\)	\(7\)

2

New Number: 5.104 | AESZ: 357 | Superseeker: 7/13 21/13 | Hash: afee0651c9b3b8e98079f5c2d5bfa8a5

Degree: 5

$13^{2} \theta^4-13 x\left(441\theta^4+690\theta^3+631\theta^2+286\theta+52\right)+2^{4} x^{2}\left(5121\theta^4+15576\theta^3+21215\theta^2+13702\theta+3445\right)-2^{10} x^{3}\left(640\theta^4+2847\theta^3+5078\theta^2+4056\theta+1196\right)+2^{14} x^{4}\left(125\theta^4+562\theta^3+905\theta^2+624\theta+157\right)-2^{21} x^{5}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 4, 20, 112, 916, ...
--> OEIS
Normalized instanton numbers (n₀=1): 7/13, -10/13, 21/13, 296/13, 608/13, ... ; Common denominator:...

Discriminant

$-(16z-1)(128z^2-13z+1)(-13+32z)^2$

Local exponents

$0$ $\frac{ 13}{ 256}-\frac{ 7}{ 256}\sqrt{ 7}I$ $\frac{ 13}{ 256}+\frac{ 7}{ 256}\sqrt{ 7}I$ $\frac{ 1}{ 16}$ $\frac{ 13}{ 32}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $1$
$0$ $1$ $1$ $1$ $1$ $1$
$0$ $1$ $1$ $1$ $3$ $1$
$0$ $2$ $2$ $2$ $4$ $1$

\(0\)	\(\frac{ 13}{ 256}-\frac{ 7}{ 256}\sqrt{ 7}I\)	\(\frac{ 13}{ 256}+\frac{ 7}{ 256}\sqrt{ 7}I\)	\(\frac{ 1}{ 16}\)	\(\frac{ 13}{ 32}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(1\)	\(1\)	\(3\)	\(1\)
\(0\)	\(2\)	\(2\)	\(2\)	\(4\)	\(1\)

Note:

There is a second MUM-point at infinity,
corresponding to Operator AESZ 358/5.105

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-128\)	\(160\)	\(208\)	\(48\)

3

New Number: 5.37 | AESZ: 221 | Superseeker: 492/5 872164/5 | Hash: b7ce7a734c057660ce3d6341a7572078

Degree: 5

$5^{2} \theta^4-2^{2} 5 x\left(404\theta^4+1096\theta^3+773\theta^2+225\theta+25\right)-2^{4} x^{2}\left(66896\theta^4+137408\theta^3+101096\theta^2+52800\theta+11625\right)-2^{8} 3 5 x^{3}(2\theta+1)(5672\theta^3+9500\theta^2+8422\theta+2689)-2^{15} 3^{2} x^{4}(2\theta+1)(1208\theta^3+2892\theta^2+2842\theta+969)-2^{20} 3^{3} x^{5}(2\theta+1)(6\theta+5)(6\theta+7)(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 20, 2988, 618320, 156299500, ...
--> OEIS
Normalized instanton numbers (n₀=1): 492/5, 10376/5, 872164/5, 91316176/5, 12181916784/5, ... ; Common denominator:...

Discriminant

$-(-1+432z)(16z+1)^2(192z+5)^2$

Local exponents

$-\frac{ 1}{ 16}$ $-\frac{ 5}{ 192}$ $0$ $\frac{ 1}{ 432}$ $\infty$
$0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$\frac{ 1}{ 4}$ $1$ $0$ $1$ $\frac{ 5}{ 6}$
$\frac{ 3}{ 4}$ $3$ $0$ $1$ $\frac{ 7}{ 6}$
$1$ $4$ $0$ $2$ $\frac{ 3}{ 2}$

\(-\frac{ 1}{ 16}\)	\(-\frac{ 5}{ 192}\)	\(0\)	\(\frac{ 1}{ 432}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(\frac{ 1}{ 4}\)	\(1\)	\(0\)	\(1\)	\(\frac{ 5}{ 6}\)
\(\frac{ 3}{ 4}\)	\(3\)	\(0\)	\(1\)	\(\frac{ 7}{ 6}\)
\(1\)	\(4\)	\(0\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

This is operator "5.37" from ...

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-32\)	\(40\)	\(10\)	\(5\)

4

New Number: 11.3 | AESZ: | Superseeker: 118/91 268/13 | Hash: 082df0c6e37c18b98ea10260e3e1c195

Degree: 11

\(7^{2} 13^{2} \theta^4+7 13 x\theta(782\theta^3-1874\theta^2-1210\theta-273)-x^{2}\left(2515785\theta^4+11622522\theta^3+15227939\theta^2+9962953\theta+2649920\right)-x^{3}\left(59827597\theta^4+258678126\theta^3+432607868\theta^2+348819198\theta+110445426\right)-2 x^{4}\left(306021521\theta^4+1499440609\theta^3+2950997910\theta^2+2719866190\theta+957861945\right)-3 x^{5}\left(1254280114\theta^4+7075609686\theta^3+15834414271\theta^2+16174233521\theta+6159865002\right)-x^{6}\left(15265487382\theta^4+98210309094\theta^3+244753624741\theta^2+271941545379\theta+110147546634\right)-2 x^{7}\left(21051636001\theta^4+152243816141\theta^3+415982528557\theta^2+495914741301\theta+211134581226\right)-2 x^{8}(\theta+1)(39253400626\theta^3+275108963001\theta^2+654332416678\theta+521254338620)-x^{9}(\theta+1)(\theta+2)(94987355417\theta^2+545340710193\theta+799002779040)-2^{2} 5 7 11 x^{10}(\theta+3)(\theta+2)(\theta+1)(43765159\theta+149264765)-2^{2} 3 5^{2} 7^{2} 11^{2} 11971 x^{11}(\theta+1)(\theta+2)(\theta+3)(\theta+4)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 20, 186, 2940, ...
--> OEIS
Normalized instanton numbers (n₀=1): 118/91, 373/91, 268/13, 12732/91, 105020/91, ... ; Common denominator:...

Discriminant

$-(3z+1)(11971z^6+16085z^5+8704z^4+2334z^3+289z^2+7z-1)(91+573z+770z^2)^2$

Local exponents

$-\frac{ 573}{ 1540}-\frac{ 1}{ 1540}\sqrt{ 48049}$ $-\frac{ 1}{ 3}$ $-\frac{ 573}{ 1540}+\frac{ 1}{ 1540}\sqrt{ 48049}$ $0$ $#ND+#NDI$ $\infty$
$0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $0$ $1$ $2$
$3$ $1$ $3$ $0$ $1$ $3$
$4$ $2$ $4$ $0$ $2$ $4$

\(-\frac{ 573}{ 1540}-\frac{ 1}{ 1540}\sqrt{ 48049}\)	\(-\frac{ 1}{ 3}\)	\(-\frac{ 573}{ 1540}+\frac{ 1}{ 1540}\sqrt{ 48049}\)	\(0\)	\(#ND+#NDI\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(0\)	\(1\)	\(2\)
\(3\)	\(1\)	\(3\)	\(0\)	\(1\)	\(3\)
\(4\)	\(2\)	\(4\)	\(0\)	\(2\)	\(4\)

Note:

This is operator "11.3" from ...

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

5

New Number: 15.3 | AESZ: | Superseeker: 44/3 220588/81 | Hash: ae51313cd958206bb1b7a3c8ae23e509

Degree: 15

\(3^{3} \theta^4+2^{2} 3^{2} x\left(12\theta^4-160\theta^3-153\theta^2-73\theta-15\right)-2^{4} 3 x^{2}\left(2688\theta^4+704\theta^3-6380\theta^2-6164\theta-2343\right)+2^{8} x^{3}\left(1312\theta^4+69632\theta^3+26456\theta^2+3928\theta-4305\right)+2^{12} x^{4}\left(51264\theta^4-16512\theta^3-16360\theta^2-16088\theta-1785\right)-2^{16} x^{5}\left(52000\theta^4+223680\theta^3+316652\theta^2+308700\theta+133179\right)-2^{21} x^{6}\left(42088\theta^4+36416\theta^3+31682\theta^2-15530\theta-24313\right)+2^{25} x^{7}\left(58136\theta^4+309440\theta^3+666728\theta^2+761160\theta+351769\right)+2^{29} x^{8}\left(30776\theta^4+26112\theta^3-81496\theta^2-231912\theta-165231\right)-2^{33} 3 x^{9}\left(16632\theta^4+120704\theta^3+332890\theta^2+441546\theta+227145\right)-2^{36} x^{10}\left(31968\theta^4+33600\theta^3-297916\theta^2-852260\theta-648637\right)+2^{40} x^{11}\left(40000\theta^4+381696\theta^3+1258584\theta^2+1813272\theta+964287\right)+2^{44} x^{12}\left(14240\theta^4+66688\theta^3+44952\theta^2-163928\theta-198345\right)-2^{48} x^{13}\left(5824\theta^4+76480\theta^3+307828\theta^2+490020\theta+272659\right)-2^{54} 5 x^{14}\left(164\theta^4+1536\theta^3+5043\theta^2+7113\theta+3693\right)-2^{60} 5^{2} x^{15}\left((\theta+2)^4\right)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 20, 388, 7344, 141636, ...
--> OEIS
Normalized instanton numbers (n₀=1): 44/3, -1421/9, 220588/81, -14752264/243, 1138508000/729, ... ; Common denominator:...

Discriminant

$-(1+16z)(1280z^2-32z-1)^2(256z^2+16z-1)^2(16z+3)^3(16z-1)^3$

Local exponents

$-\frac{ 3}{ 16}$ $-\frac{ 1}{ 32}-\frac{ 1}{ 32}\sqrt{ 5}$ $-\frac{ 1}{ 16}$ $\frac{ 1}{ 80}-\frac{ 1}{ 80}\sqrt{ 6}$ $0$ $-\frac{ 1}{ 32}+\frac{ 1}{ 32}\sqrt{ 5}$ $\frac{ 1}{ 80}+\frac{ 1}{ 80}\sqrt{ 6}$ $\frac{ 1}{ 16}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $2$
$2$ $\frac{ 1}{ 2}$ $1$ $1$ $0$ $\frac{ 1}{ 2}$ $1$ $0$ $2$
$3$ $\frac{ 1}{ 2}$ $1$ $3$ $0$ $\frac{ 1}{ 2}$ $3$ $0$ $2$
$5$ $1$ $2$ $4$ $0$ $1$ $4$ $0$ $2$

\(-\frac{ 3}{ 16}\)	\(-\frac{ 1}{ 32}-\frac{ 1}{ 32}\sqrt{ 5}\)	\(-\frac{ 1}{ 16}\)	\(\frac{ 1}{ 80}-\frac{ 1}{ 80}\sqrt{ 6}\)	\(0\)	\(-\frac{ 1}{ 32}+\frac{ 1}{ 32}\sqrt{ 5}\)	\(\frac{ 1}{ 80}+\frac{ 1}{ 80}\sqrt{ 6}\)	\(\frac{ 1}{ 16}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(2\)
\(2\)	\(\frac{ 1}{ 2}\)	\(1\)	\(1\)	\(0\)	\(\frac{ 1}{ 2}\)	\(1\)	\(0\)	\(2\)
\(3\)	\(\frac{ 1}{ 2}\)	\(1\)	\(3\)	\(0\)	\(\frac{ 1}{ 2}\)	\(3\)	\(0\)	\(2\)
\(5\)	\(1\)	\(2\)	\(4\)	\(0\)	\(1\)	\(4\)	\(0\)	\(2\)

Note:

This is operator "15.3" from ...

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

6

New Number: 6.8 | AESZ: | Superseeker: 567/13 512341/13 | Hash: 00104510dfaa4ae75940f08df0a52bf5

Degree: 6

$13^{2} \theta^4-13 x\left(5041\theta^4+7634\theta^3+5767\theta^2+1950\theta+260\right)+2^{3} x^{2}\left(744635\theta^4+1560842\theta^3+1510101\theta^2+768170\theta+156078\right)-2^{6} 3 x^{3}\left(1232985\theta^4+3409302\theta^3+4189688\theta^2+2419209\theta+518414\right)+2^{9} x^{4}\left(9225025\theta^4+33675338\theta^3+49289090\theta^2+31849807\theta+7296732\right)-2^{12} 3 17 x^{5}(\theta+1)(222704\theta^3+833160\theta^2+989659\theta+317310)+2^{15} 3^{2} 17^{2} 23^{2} x^{6}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 20, 1524, 196400, 31587220, ...
--> OEIS
Normalized instanton numbers (n₀=1): 567/13, 11392/13, 512341/13, 34191454/13, 2850663840/13, ... ; Common denominator:...

Discriminant

$(1-293z+4232z^2)(408z-13)^2(16z-1)^2$

Local exponents

$0$ $\frac{ 293}{ 8464}-\frac{ 41}{ 8464}\sqrt{ 41}$ $\frac{ 13}{ 408}$ $\frac{ 1}{ 16}$ $\frac{ 293}{ 8464}+\frac{ 41}{ 8464}\sqrt{ 41}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $\frac{ 1}{ 2}$ $1$ $1$
$0$ $1$ $3$ $\frac{ 1}{ 2}$ $1$ $2$
$0$ $2$ $4$ $1$ $2$ $\frac{ 5}{ 2}$

\(0\)	\(\frac{ 293}{ 8464}-\frac{ 41}{ 8464}\sqrt{ 41}\)	\(\frac{ 13}{ 408}\)	\(\frac{ 1}{ 16}\)	\(\frac{ 293}{ 8464}+\frac{ 41}{ 8464}\sqrt{ 41}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 1}{ 2}\)	\(1\)	\(1\)
\(0\)	\(1\)	\(3\)	\(\frac{ 1}{ 2}\)	\(1\)	\(2\)
\(0\)	\(2\)	\(4\)	\(1\)	\(2\)	\(\frac{ 5}{ 2}\)

Note:

This is operator "6.8" from ...

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

7

New Number: 8.40 | AESZ: | Superseeker: 5/4 35/2 | Hash: d3cb0fbbc65d6c5dace733d3d1ca181b

Degree: 8

$2^{4} \theta^4-2^{2} x\theta(2\theta^3+82\theta^2+53\theta+12)-x^{2}\left(4895\theta^4+18410\theta^3+26199\theta^2+18308\theta+5120\right)-x^{3}\left(60679\theta^4+272424\theta^3+497452\theta^2+430092\theta+143808\right)-x^{4}\left(344527\theta^4+1870838\theta^3+4034628\theta^2+3987101\theta+1478544\right)-x^{5}(\theta+1)(1076509\theta^3+5847783\theta^2+11226106\theta+7492832)-2 x^{6}(\theta+1)(\theta+2)(944887\theta^2+4249317\theta+5045304)-2^{8} 13 x^{7}(\theta+3)(\theta+2)(\theta+1)(518\theta+1381)-2^{5} 5 13^{2} 23 x^{8}(\theta+1)(\theta+2)(\theta+3)(\theta+4)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 20, 168, 2652, ...
--> OEIS
Normalized instanton numbers (n₀=1): 5/4, 57/16, 35/2, 459/4, 3615/4, ... ; Common denominator:...

Discriminant

$-(23z-1)(5z+1)(2z+1)(z+1)(13z+4)^2(4z+1)^2$

Local exponents

$-1$ $-\frac{ 1}{ 2}$ $-\frac{ 4}{ 13}$ $-\frac{ 1}{ 4}$ $-\frac{ 1}{ 5}$ $0$ $\frac{ 1}{ 23}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $1$ $\frac{ 1}{ 3}$ $1$ $0$ $1$ $2$
$1$ $1$ $3$ $\frac{ 2}{ 3}$ $1$ $0$ $1$ $3$
$2$ $2$ $4$ $1$ $2$ $0$ $2$ $4$

\(-1\)	\(-\frac{ 1}{ 2}\)	\(-\frac{ 4}{ 13}\)	\(-\frac{ 1}{ 4}\)	\(-\frac{ 1}{ 5}\)	\(0\)	\(\frac{ 1}{ 23}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(1\)	\(\frac{ 1}{ 3}\)	\(1\)	\(0\)	\(1\)	\(2\)
\(1\)	\(1\)	\(3\)	\(\frac{ 2}{ 3}\)	\(1\)	\(0\)	\(1\)	\(3\)
\(2\)	\(2\)	\(4\)	\(1\)	\(2\)	\(0\)	\(2\)	\(4\)

Note:

This is operator "8.40" 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:

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