Calabi-Yau differential operator database v.3.0

1

New Number: 2.67 | AESZ: 245 | Superseeker: -6 -170 | Hash: 0ef0bca0dbecbedad92696fd7c0f9e42

Degree: 2

$\theta^4-2 3 x\left(36\theta^4+66\theta^3+61\theta^2+28\theta+5\right)+2^{2} 3^{2} x^{2}(3\theta+2)^2(6\theta+7)^2$

Coefficients of the holomorphic solution: 1, 30, 1764, 127776, 10248750, ...
--> OEIS
Normalized instanton numbers (n₀=1): -6, -33, -170, -1029, -3246, ... ; Common denominator:...

Discriminant

$(108z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 108}$ $\infty$
$0$ $0$ $\frac{ 2}{ 3}$
$0$ $\frac{ 1}{ 6}$ $\frac{ 2}{ 3}$
$0$ $1$ $\frac{ 7}{ 6}$
$0$ $\frac{ 7}{ 6}$ $\frac{ 7}{ 6}$

\(0\)	\(\frac{ 1}{ 108}\)	\(\infty\)
\(0\)	\(0\)	\(\frac{ 2}{ 3}\)
\(0\)	\(\frac{ 1}{ 6}\)	\(\frac{ 2}{ 3}\)
\(0\)	\(1\)	\(\frac{ 7}{ 6}\)
\(0\)	\(\frac{ 7}{ 6}\)	\(\frac{ 7}{ 6}\)

Note:

This is operator "2.67" from ...

Integral instantons: ,...
Coefficients of the Yukawa coupling: 1, -6, -270, -4596, -66126, -405756, 8587836, 424349304,...
Coefficients of the q-coordinate : 0, 1, -48, 1440, -32848, 637872, -11009952, 174295160,...

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

No topological data

Monodromy (with respect to Frobenius basis)

$1$	$-1$	$\frac{ 1}{ 2}$	$-\frac{ 1}{ 6}$
$0$	$1$	$-1$	$\frac{ 1}{ 2}$
$0$	$0$	$1$	$-1$
$0$	$0$	$0$	$1$

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$1.052168056+.90357723e-1I$	$\frac{ 1}{ 4}-\frac{ 1}{ 12}I\sqrt{ 3}$	$-33^{ \frac{ 1}{ 6}}Pi^{ \frac{ 9}{ 2}}Zeta(5)^6-6\lambda$	$.10688274e-1-\frac{ 1}{ 5}I5^{ \frac{ 4}{ 7}}ln(2)^12$
$\frac{ 3}{ 2}-\frac{ 1}{ 2}I\sqrt{ 3}$	$\frac{ 3}{ 4}-\frac{ 1}{ 4}I\sqrt{ 3}$	$-\frac{ 1}{ 8}+\frac{ 1}{ 24}I\sqrt{ 3}$	$33^{ \frac{ 1}{ 6}}Pi^{ \frac{ 9}{ 2}}Zeta(5)^6+6\lambda$
$0$	$\frac{ 9}{ 2}-\frac{ 3}{ 2}I\sqrt{ 3}$	$\frac{ 3}{ 4}-\frac{ 1}{ 4}I\sqrt{ 3}$	$\frac{ 1}{ 4}-\frac{ 1}{ 12}I\sqrt{ 3}$
$-27+9I\sqrt{ 3}$	$0$	$\frac{ 3}{ 2}-\frac{ 1}{ 2}I\sqrt{ 3}$	$.447831944-.956383127I$

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Basis of the Doran-Morgan lattice

$-\frac{ 6521007}{ 125000000}-\frac{ 90357723}{ 1000000000}I$	$-3+\frac{ 4511615154067544842086723740124334848694351954431101272196462779644508392806538050844596873167533211}{ 2604782223680891931509451724762066519358428478691519147090699072692122550847025398588836869330367971}I$	$\frac{ 1}{ 2}-\frac{ 3081490456277555159153769728602633601692409347626414678367139999430219011336903561652776870289659281}{ 3558198688874218386798087732443200684026390216561310209643580926168315471826781724716716871248950591}I$	$1$
$-\frac{ 3}{ 2}+\frac{ 1}{ 2}I\sqrt{ 3}$	$\frac{ 27}{ 2}-\frac{ 5941739851857534525019677751646036095696294561235787866025785559858797774276172540036416876045407141}{ 762325578427705482137746574067229488156369299800529597635468373844463859062368874404292289563613477}I$	$-\frac{ 1}{ 2}+\frac{ 3081490456277555159153769728602633601692409347626414678367139999430219011336903561652776870289659281}{ 3558198688874218386798087732443200684026390216561310209643580926168315471826781724716716871248950591}I$	$0$
$0$	$-27+\frac{ 11883479703715069050039355503292072191392589122471575732051571119717595548552345080072833752090814282}{ 762325578427705482137746574067229488156369299800529597635468373844463859062368874404292289563613477}I$	$0$	$0$
$27-\frac{ 11883479703715069050039355503292072191392589122471575732051571119717595548552345080072833752090814282}{ 762325578427705482137746574067229488156369299800529597635468373844463859062368874404292289563613477}I$	$0$	$0$	$0$

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2

New Number: 4.25 | AESZ: 32 | Superseeker: -33 -13051 | Hash: bf53401dcbe0436fb67761f590ee3295

Degree: 4

$\theta^4-x\left(540\theta^4+1080\theta^3+1296\theta^2+756\theta+339/2\right)+x^{2}\left(72846\theta^4+291384\theta^3+881067/2\theta^2+298299\theta+305217/4\right)+x^{3}\left(14580\theta^4+87480\theta^3+209547\theta^2+234981\theta+205497/2\right)+x^{4}9/16(6\theta+11)^2(6\theta+13)^2$

Maple LaTex

Coefficients of the holomorphic solution: 1, 339/2, 287415/8, 131845323/16, 251852894379/128, ...
--> OEIS
Normalized instanton numbers (n₀=1): -33, -1995/4, -13051, -435975, -16838124, ... ; Common denominator:...

Discriminant

$(-1+270z+27z^2)^2$

Local exponents

$-5-\frac{ 26}{ 9}\sqrt{ 3}$ $0$ $s_1$ $s_2$ $-5+\frac{ 26}{ 9}\sqrt{ 3}$ $\infty$
$0$ $0$ $-\frac{ 1}{ 2}$ $-\frac{ 1}{ 2}$ $0$ $\frac{ 11}{ 6}$
$-\frac{ 1}{ 2}$ $0$ $0$ $0$ $-\frac{ 1}{ 2}$ $\frac{ 11}{ 6}$
$1$ $0$ $1$ $1$ $1$ $\frac{ 13}{ 6}$
$\frac{ 3}{ 2}$ $0$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 13}{ 6}$

\(-5-\frac{ 26}{ 9}\sqrt{ 3}\)	\(0\)	\(s_1\)	\(s_2\)	\(-5+\frac{ 26}{ 9}\sqrt{ 3}\)	\(\infty\)
\(0\)	\(0\)	\(-\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 2}\)	\(0\)	\(\frac{ 11}{ 6}\)
\(-\frac{ 1}{ 2}\)	\(0\)	\(0\)	\(0\)	\(-\frac{ 1}{ 2}\)	\(\frac{ 11}{ 6}\)
\(1\)	\(0\)	\(1\)	\(1\)	\(1\)	\(\frac{ 13}{ 6}\)
\(\frac{ 3}{ 2}\)	\(0\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 13}{ 6}\)

Note:

Sporadic YY-Operator

3

New Number: 4.9 | AESZ: | Superseeker: -33 29693 | Hash: c444fb1a912bd488ee5947b8bc1e2c53

Degree: 4

$\theta^4-x\left(756\theta^4+1512\theta^3+1824\theta^2+1068\theta+483/2\right)+x^{2}\left(260982\theta^4+1043928\theta^3+3947211/2\theta^2+1859355\theta+2817729/4\right)-x^{3}531441/2(2\theta+3)^2(42\theta^2+126\theta+127)+x^{4}387420489/16(2\theta+3)(2\theta+5)(6\theta+11)(6\theta+13)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 483/2, 300015/8, 32162403/16, -132658029189/128, ...
--> OEIS
Normalized instanton numbers (n₀=1): -33, 1095/2, 29693, 1241103/2, -16117818, ... ; Common denominator:...

Discriminant

$(1-378z+59049z^2)^2$

Local exponents

$0$ $s_1$ $s_2$ $\frac{ 7}{ 2187}-\frac{ 4}{ 2187}\sqrt{ 2}I$ $\frac{ 7}{ 2187}+\frac{ 4}{ 2187}\sqrt{ 2}I$ $\infty$
$0$ $-\frac{ 1}{ 2}$ $-\frac{ 1}{ 2}$ $0$ $0$ $\frac{ 3}{ 2}$
$0$ $0$ $0$ $-\frac{ 1}{ 2}$ $-\frac{ 1}{ 2}$ $\frac{ 11}{ 6}$
$0$ $1$ $1$ $1$ $1$ $\frac{ 13}{ 6}$
$0$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$ $\frac{ 5}{ 2}$

\(0\)	\(s_1\)	\(s_2\)	\(\frac{ 7}{ 2187}-\frac{ 4}{ 2187}\sqrt{ 2}I\)	\(\frac{ 7}{ 2187}+\frac{ 4}{ 2187}\sqrt{ 2}I\)	\(\infty\)
\(0\)	\(-\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 2}\)	\(0\)	\(0\)	\(\frac{ 3}{ 2}\)
\(0\)	\(0\)	\(0\)	\(-\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 2}\)	\(\frac{ 11}{ 6}\)
\(0\)	\(1\)	\(1\)	\(1\)	\(1\)	\(\frac{ 13}{ 6}\)
\(0\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)	\(\frac{ 5}{ 2}\)

Note:

YY-operator equivalent to AESZ 151=$B \ast \delta$

\(1.052168056+.90357723e-1I\)	\(\frac{ 1}{ 4}-\frac{ 1}{ 12}I\sqrt{ 3}\)	\(-33^{ \frac{ 1}{ 6}}Pi^{ \frac{ 9}{ 2}}Zeta(5)^6-6\lambda\)	\(.10688274e-1-\frac{ 1}{ 5}I5^{ \frac{ 4}{ 7}}ln(2)^12\)
\(\frac{ 3}{ 2}-\frac{ 1}{ 2}I\sqrt{ 3}\)	\(\frac{ 3}{ 4}-\frac{ 1}{ 4}I\sqrt{ 3}\)	\(-\frac{ 1}{ 8}+\frac{ 1}{ 24}I\sqrt{ 3}\)	\(33^{ \frac{ 1}{ 6}}Pi^{ \frac{ 9}{ 2}}Zeta(5)^6+6\lambda\)
\(0\)	\(\frac{ 9}{ 2}-\frac{ 3}{ 2}I\sqrt{ 3}\)	\(\frac{ 3}{ 4}-\frac{ 1}{ 4}I\sqrt{ 3}\)	\(\frac{ 1}{ 4}-\frac{ 1}{ 12}I\sqrt{ 3}\)
\(-27+9I\sqrt{ 3}\)	\(0\)	\(\frac{ 3}{ 2}-\frac{ 1}{ 2}I\sqrt{ 3}\)	\(.447831944-.956383127I\)

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

\(-\frac{ 6521007}{ 125000000}-\frac{ 90357723}{ 1000000000}I\)	\(-3+\frac{ 4511615154067544842086723740124334848694351954431101272196462779644508392806538050844596873167533211}{ 2604782223680891931509451724762066519358428478691519147090699072692122550847025398588836869330367971}I\)	\(\frac{ 1}{ 2}-\frac{ 3081490456277555159153769728602633601692409347626414678367139999430219011336903561652776870289659281}{ 3558198688874218386798087732443200684026390216561310209643580926168315471826781724716716871248950591}I\)	\(1\)
\(-\frac{ 3}{ 2}+\frac{ 1}{ 2}I\sqrt{ 3}\)	\(\frac{ 27}{ 2}-\frac{ 5941739851857534525019677751646036095696294561235787866025785559858797774276172540036416876045407141}{ 762325578427705482137746574067229488156369299800529597635468373844463859062368874404292289563613477}I\)	\(-\frac{ 1}{ 2}+\frac{ 3081490456277555159153769728602633601692409347626414678367139999430219011336903561652776870289659281}{ 3558198688874218386798087732443200684026390216561310209643580926168315471826781724716716871248950591}I\)	\(0\)
\(0\)	\(-27+\frac{ 11883479703715069050039355503292072191392589122471575732051571119717595548552345080072833752090814282}{ 762325578427705482137746574067229488156369299800529597635468373844463859062368874404292289563613477}I\)	\(0\)	\(0\)
\(27-\frac{ 11883479703715069050039355503292072191392589122471575732051571119717595548552345080072833752090814282}{ 762325578427705482137746574067229488156369299800529597635468373844463859062368874404292289563613477}I\)	\(0\)	\(0\)	\(0\)

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