Calabi-Yau differential operator database v.3.0

1

New Number: 3.27 | AESZ: 408 | Superseeker: -60 -61780 | Hash: 32ab77c73baf49023973ad11e5d0852e

Degree: 3

$\theta^4-2^{2} x(2\theta+1)(46\theta^3+53\theta^2+45\theta+11)-2^{4} x^{2}(8\theta+7)(64\theta^3+312\theta^2+440\theta+135)+2^{8} 3^{2} x^{3}(8\theta+3)(8\theta+7)(8\theta+15)(8\theta+19)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 44, 6060, 972720, 182017260, ...
--> OEIS
Normalized instanton numbers (n₀=1): -60, 975, -61780, 4166460, -1853578608/5, ... ; Common denominator:...

Discriminant

$(144z+1)(-1+256z)^2$

Local exponents

$-\frac{ 1}{ 144}$ $0$ $\frac{ 1}{ 256}$ $\infty$
$0$ $0$ $0$ $\frac{ 3}{ 8}$
$1$ $0$ $\frac{ 1}{ 4}$ $\frac{ 7}{ 8}$
$1$ $0$ $1$ $\frac{ 15}{ 8}$
$2$ $0$ $\frac{ 5}{ 4}$ $\frac{ 19}{ 8}$

\(-\frac{ 1}{ 144}\)	\(0\)	\(\frac{ 1}{ 256}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 3}{ 8}\)
\(1\)	\(0\)	\(\frac{ 1}{ 4}\)	\(\frac{ 7}{ 8}\)
\(1\)	\(0\)	\(1\)	\(\frac{ 15}{ 8}\)
\(2\)	\(0\)	\(\frac{ 5}{ 4}\)	\(\frac{ 19}{ 8}\)

Note:

This is operator "3.27" from ...

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

2

New Number: 5.125 | AESZ: | Superseeker: 229 10128562/3 | Hash: ce147b64cc67dee1204a2d16f6ac4210

Degree: 5

$\theta^4-x\left(1217\theta^4+2050\theta^3+1437\theta^2+412\theta+44\right)+2^{5} x^{2}(\theta+1)(4550\theta^3-186\theta^2-899\theta-171)-2^{8} x^{3}\left(18484\theta^4+3192\theta^3+1005\theta^2+1107\theta+258\right)+2^{14} x^{4}(2\theta+1)(268\theta^3+414\theta^2+267\theta+65)-2^{20} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 44, 14532, 7508960, 4749338020, ...
--> OEIS
Normalized instanton numbers (n₀=1): 229, 18542, 10128562/3, 938391582, 323686899951, ... ; Common denominator:...

Discriminant

$-(z-1)(1024z^2-1088z+1)(-1+64z)^2$

Local exponents

$0$ $\frac{ 17}{ 32}-\frac{ 3}{ 8}\sqrt{ 2}$ $\frac{ 1}{ 64}$ $1$ $\frac{ 17}{ 32}+\frac{ 3}{ 8}\sqrt{ 2}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $\frac{ 1}{ 2}$
$0$ $1$ $1$ $1$ $1$ $1$
$0$ $1$ $3$ $1$ $1$ $1$
$0$ $2$ $4$ $2$ $2$ $\frac{ 3}{ 2}$

\(0\)	\(\frac{ 17}{ 32}-\frac{ 3}{ 8}\sqrt{ 2}\)	\(\frac{ 1}{ 64}\)	\(1\)	\(\frac{ 17}{ 32}+\frac{ 3}{ 8}\sqrt{ 2}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 2}\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(1\)
\(0\)	\(1\)	\(3\)	\(1\)	\(1\)	\(1\)
\(0\)	\(2\)	\(4\)	\(2\)	\(2\)	\(\frac{ 3}{ 2}\)

Note:

B-Incarnation as fibre product 62211- x 362--1

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

3

New Number: 5.47 | AESZ: 246 | Superseeker: -4/5 -108/5 | Hash: f51a0c39f9179dc6a561b9afb6f9d85f

Degree: 5

$5^{2} \theta^4-2^{2} 5 x\left(12\theta^4+48\theta^3+49\theta^2+25\theta+5\right)-2^{4} x^{2}\left(544\theta^4+1792\theta^3+2444\theta^2+1580\theta+405\right)+2^{9} x^{3}\left(112\theta^4+960\theta^3+2306\theta^2+2130\theta+685\right)+2^{12} x^{4}\left(144\theta^4+768\theta^3+1308\theta^2+924\theta+235\right)+2^{20} x^{5}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 4, 44, 400, 5356, ...
--> OEIS
Normalized instanton numbers (n₀=1): -4/5, 22/5, -108/5, 694/5, -1040, ... ; Common denominator:...

Discriminant

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

Local exponents

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

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

Note:

There is a second MUM-point at infinity,
corresponding to Operator AESZ 247/5.48

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

4

New Number: 5.4 | AESZ: 21 | Superseeker: 8/5 152/5 | Hash: 42a2bc0f0ee2a405ede956176c95721f

Degree: 5

$5^{2} \theta^4-2^{2} 5 x\left(36\theta^4+84\theta^3+72\theta^2+30\theta+5\right)-2^{4} x^{2}\left(181\theta^4+268\theta^3+71\theta^2-70\theta-35\right)+2^{8} x^{3}(\theta+1)(37\theta^3+248\theta^2+375\theta+165)+2^{10} x^{4}\left(39\theta^4+198\theta^3+331\theta^2+232\theta+59\right)+2^{15} x^{5}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 4, 44, 688, 13036, ...
--> OEIS
Normalized instanton numbers (n₀=1): 8/5, 57/10, 152/5, 253, 11552/5, ... ; Common denominator:...

Discriminant

$(4z+1)(32z-1)(4z-1)(8z+5)^2$

Local exponents

$-\frac{ 5}{ 8}$ $-\frac{ 1}{ 4}$ $0$ $\frac{ 1}{ 32}$ $\frac{ 1}{ 4}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $1$
$1$ $1$ $0$ $1$ $1$ $1$
$3$ $1$ $0$ $1$ $1$ $1$
$4$ $2$ $0$ $2$ $2$ $1$

\(-\frac{ 5}{ 8}\)	\(-\frac{ 1}{ 4}\)	\(0\)	\(\frac{ 1}{ 32}\)	\(\frac{ 1}{ 4}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(3\)	\(1\)	\(0\)	\(1\)	\(1\)	\(1\)
\(4\)	\(2\)	\(0\)	\(2\)	\(2\)	\(1\)

Note:

There is a second MUM-point at infinity, corresponding to
Operator AESZ 71/5.11

A-Incarnation: (2,0),(02),(1,1),(1,1),(1,1) intersection in $P^4 \times P^4$

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-88\)	\(104\)	\(80\)	\(22\)

5

New Number: 15.4 | AESZ: | Superseeker: 52/5 13436/5 | Hash: 2306e85a3af0a97d616dedf03cc93f69

Degree: 15

\(5^{2} \theta^4-2^{2} 5 x\left(524\theta^4+56\theta^3+83\theta^2+55\theta+15\right)+2^{4} x^{2}\left(122784\theta^4+39552\theta^3+60584\theta^2+42560\theta+9895\right)-2^{8} x^{3}\left(851424\theta^4+544704\theta^3+819724\theta^2+563860\theta+144605\right)+2^{13} x^{4}\left(1949840\theta^4+2047744\theta^3+3062224\theta^2+2155304\theta+617905\right)-2^{18} x^{5}\left(3117952\theta^4+4806720\theta^3+7335648\theta^2+5468420\theta+1717063\right)+2^{22} x^{6}\left(7179524\theta^4+15086448\theta^3+24112808\theta^2+19319920\theta+6533401\right)-2^{26} x^{7}\left(12098492\theta^4+32868584\theta^3+56087648\theta^2+48438116\theta+17467537\right)+2^{31} x^{8}\left(7508036\theta^4+25345280\theta^3+46719420\theta^2+43397656\theta+16591239\right)-2^{38} x^{9}\left(856369\theta^4+3481940\theta^3+6970670\theta^2+6938899\theta+2800514\right)+2^{42} x^{10}\left(568775\theta^4+2715196\theta^3+5906890\theta^2+6274274\theta+2662654\right)-2^{46} x^{11}\left(269591\theta^4+1478382\theta^3+3484287\theta^2+3929620\theta+1745534\right)+2^{51} x^{12}\left(44091\theta^4+272424\theta^3+691403\theta^2+822862\theta+380404\right)-2^{57} x^{13}\left(2349\theta^4+16068\theta^3+43548\theta^2+54271\theta+25924\right)+2^{63} x^{14}\left(73\theta^4+544\theta^3+1559\theta^2+2017\theta+988\right)-2^{69} x^{15}\left((\theta+2)^4\right)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 44, -3792, -207124, ...
--> OEIS
Normalized instanton numbers (n₀=1): 52/5, 115, 13436/5, 89632, 18465296/5, ... ; Common denominator:...

Discriminant

$-(-1+32z)(256z^2-48z+1)^2(512z^2-128z+5)^2(64z-1)^3(16z-1)^3$

Local exponents

$0$ $\frac{ 1}{ 64}$ $\frac{ 3}{ 32}-\frac{ 1}{ 32}\sqrt{ 5}$ $\frac{ 1}{ 32}$ $\frac{ 1}{ 8}-\frac{ 1}{ 32}\sqrt{ 6}$ $\frac{ 1}{ 16}$ $\frac{ 3}{ 32}+\frac{ 1}{ 32}\sqrt{ 5}$ $\frac{ 1}{ 8}+\frac{ 1}{ 32}\sqrt{ 6}$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$ $2$
$0$ $2$ $\frac{ 1}{ 2}$ $1$ $1$ $0$ $\frac{ 1}{ 2}$ $1$ $2$
$0$ $3$ $\frac{ 1}{ 2}$ $1$ $3$ $0$ $\frac{ 1}{ 2}$ $3$ $2$
$0$ $5$ $1$ $2$ $4$ $0$ $1$ $4$ $2$

\(0\)	\(\frac{ 1}{ 64}\)	\(\frac{ 3}{ 32}-\frac{ 1}{ 32}\sqrt{ 5}\)	\(\frac{ 1}{ 32}\)	\(\frac{ 1}{ 8}-\frac{ 1}{ 32}\sqrt{ 6}\)	\(\frac{ 1}{ 16}\)	\(\frac{ 3}{ 32}+\frac{ 1}{ 32}\sqrt{ 5}\)	\(\frac{ 1}{ 8}+\frac{ 1}{ 32}\sqrt{ 6}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(2\)
\(0\)	\(2\)	\(\frac{ 1}{ 2}\)	\(1\)	\(1\)	\(0\)	\(\frac{ 1}{ 2}\)	\(1\)	\(2\)
\(0\)	\(3\)	\(\frac{ 1}{ 2}\)	\(1\)	\(3\)	\(0\)	\(\frac{ 1}{ 2}\)	\(3\)	\(2\)
\(0\)	\(5\)	\(1\)	\(2\)	\(4\)	\(0\)	\(1\)	\(4\)	\(2\)

Note:

This is operator "15.4" 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:

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