Calabi-Yau differential operator database v.3.0

1

New Number: 3.12 | AESZ: | Superseeker: 252 1162036 | Hash: baa148eb1a5a05a0d9aca4c78be26905

Degree: 3

$\theta^4-2^{2} 3^{2} x\left(132\theta^4+216\theta^3+165\theta^2+57\theta+7\right)+2^{4} 3^{6} x^{2}(4\theta+3)(160\theta^3+408\theta^2+316\theta+57)-2^{8} 3^{10} x^{3}(4\theta+1)(4\theta+3)(4\theta+7)(4\theta+9)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 252, 202500, 212132880, 251548748100, ...
--> OEIS
Normalized instanton numbers (n₀=1): 252, -19512, 1162036, -91851948, 24209298720, ... ; Common denominator:...

Discriminant

$-(1296z-1)(-1+1728z)^2$

Local exponents

$0$ $\frac{ 1}{ 1728}$ $\frac{ 1}{ 1296}$ $\infty$
$0$ $0$ $0$ $\frac{ 1}{ 4}$
$0$ $\frac{ 1}{ 2}$ $1$ $\frac{ 3}{ 4}$
$0$ $1$ $1$ $\frac{ 7}{ 4}$
$0$ $\frac{ 3}{ 2}$ $2$ $\frac{ 9}{ 4}$

\(0\)	\(\frac{ 1}{ 1728}\)	\(\frac{ 1}{ 1296}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(\frac{ 1}{ 4}\)
\(0\)	\(\frac{ 1}{ 2}\)	\(1\)	\(\frac{ 3}{ 4}\)
\(0\)	\(1\)	\(1\)	\(\frac{ 7}{ 4}\)
\(0\)	\(\frac{ 3}{ 2}\)	\(2\)	\(\frac{ 9}{ 4}\)

Note:

Operator equivalent to AESZ 154

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

2

New Number: 5.3 | AESZ: 20 | Superseeker: 3 245/3 | Hash: a9a698dc5c79ffda497a7897390408b0

Degree: 5

$\theta^4-3 x\left(48\theta^4+60\theta^3+53\theta^2+23\theta+4\right)+3^{2} x^{2}\left(873\theta^4+1980\theta^3+2319\theta^2+1344\theta+304\right)-2 3^{4} x^{3}\left(1269\theta^4+3888\theta^3+5259\theta^2+3348\theta+800\right)+2^{2} 3^{6} x^{4}\left(891\theta^4+3240\theta^3+4653\theta^2+2952\theta+688\right)-2^{3} 3^{11} x^{5}(\theta+1)^2(3\theta+2)(3\theta+4)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 12, 252, 6600, 198540, ...
--> OEIS
Normalized instanton numbers (n₀=1): 3, 33/2, 245/3, 879, 11829, ... ; Common denominator:...

Discriminant

$-(54z-1)(27z-1)^2(18z-1)^2$

Local exponents

$0$ $\frac{ 1}{ 54}$ $\frac{ 1}{ 27}$ $\frac{ 1}{ 18}$ $\infty$
$0$ $0$ $0$ $0$ $\frac{ 2}{ 3}$
$0$ $1$ $\frac{ 1}{ 3}$ $1$ $1$
$0$ $1$ $\frac{ 2}{ 3}$ $3$ $1$
$0$ $2$ $1$ $4$ $\frac{ 4}{ 3}$

\(0\)	\(\frac{ 1}{ 54}\)	\(\frac{ 1}{ 27}\)	\(\frac{ 1}{ 18}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(\frac{ 2}{ 3}\)
\(0\)	\(1\)	\(\frac{ 1}{ 3}\)	\(1\)	\(1\)
\(0\)	\(1\)	\(\frac{ 2}{ 3}\)	\(3\)	\(1\)
\(0\)	\(2\)	\(1\)	\(4\)	\(\frac{ 4}{ 3}\)

Note:

A-Incarnation: (3,0),(0,3),(1,1) intersection in $P^3 \times \P^3$.

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

\(C_3\)	\(C_2 H\)	\(H^3\)	\( \vert H \vert\)
\(-18\)	\(72\)	\(54\)	\(15\)

3

New Number: 5.60 | AESZ: 268 | Superseeker: -828/5 -4270932/5 | Hash: 638e2881183378c7a47b7508d9acc072

Degree: 5

$5^{2} \theta^4-2^{2} 3 5 x\left(108\theta^4+432\theta^3+661\theta^2+445\theta+105\right)-2^{4} 3^{2} x^{2}\left(44064\theta^4+145152\theta^3+239004\theta^2+186300\theta+58045\right)+2^{9} 3^{5} x^{3}\left(9072\theta^4+77760\theta^3+180954\theta^2+164970\theta+53965\right)+2^{12} 3^{8} x^{4}\left(11664\theta^4+62208\theta^3+104940\theta^2+73836\theta+18659\right)+2^{20} 3^{15} x^{5}\left((\theta+1)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 252, 87084, 31502448, 12121584876, ...
--> OEIS
Normalized instanton numbers (n₀=1): -828/5, 25533/5, -4270932/5, 598304142/5, -24767201520, ... ; Common denominator:...

Discriminant

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

Local exponents

$-\frac{ 5}{ 432}$ $-\frac{ 1}{ 432}$ $0$ $\frac{ 1}{ 432}$ $\infty$
$0$ $0$ $0$ $0$ $1$
$1$ $1$ $0$ $-\frac{ 1}{ 2}$ $1$
$3$ $1$ $0$ $1$ $1$
$4$ $2$ $0$ $\frac{ 3}{ 2}$ $1$

\(-\frac{ 5}{ 432}\)	\(-\frac{ 1}{ 432}\)	\(0\)	\(\frac{ 1}{ 432}\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(1\)	\(0\)	\(-\frac{ 1}{ 2}\)	\(1\)
\(3\)	\(1\)	\(0\)	\(1\)	\(1\)
\(4\)	\(2\)	\(0\)	\(\frac{ 3}{ 2}\)	\(1\)

Note:

There is a second MUM-point at infinity, correspondint to
Operator AESZ 269/5.61

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

4

New Number: 14.6 | AESZ: | Superseeker: 343/26 27836/13 | Hash: b419826ae9a841fd2485cf17a33e3c82

Degree: 14

\(2^{2} 13^{2} \theta^4+2 13 x\theta(374\theta^3-3806\theta^2-2423\theta-520)-3 x^{2}\left(1378165\theta^4+5692942\theta^3+6262109\theta^2+3632408\theta+908544\right)-3 x^{3}\left(109634670\theta^4+414665370\theta^3+585954355\theta^2+424721388\theta+125838648\right)-3^{2} x^{4}\left(1430057388\theta^4+5835126030\theta^3+9693559559\theta^2+7980072398\theta+2602285652\right)-3^{4} x^{5}\left(3973724102\theta^4+18016019762\theta^3+33809871817\theta^2+30548046888\theta+10682005352\right)-3^{5} x^{6}\left(23181342780\theta^4+117157350210\theta^3+242997310916\theta^2+236792965009\theta+87556639706\right)-3^{6} x^{7}\left(98661453307\theta^4+553704139946\theta^3+1252095727942\theta^2+1301834765069\theta+504487460698\right)-3^{7} x^{8}\left(312059119661\theta^4+1933538622170\theta^3+4722871403800\theta^2+5200539067181\theta+2098928967026\right)-3^{7} x^{9}\left(2207453009832\theta^4+15008943280014\theta^3+39332490555167\theta^2+45616623444051\theta+19085482478826\right)-3^{8} x^{10}\left(3839323955127\theta^4+28479004361040\theta^3+79653137355055\theta^2+96880903986262\theta+41867518152496\right)-3^{10} x^{11}(\theta+1)(1597618239529\theta^3+11261457267015\theta^2+26962321719782\theta+21625438724040)-3^{12} x^{12}(\theta+1)(\theta+2)(452183900223\theta^2+2573271558279\theta+3747021993116)-2^{4} 3^{16} 109 x^{13}(\theta+3)(\theta+2)(\theta+1)(4973417\theta+16619273)-2^{6} 3^{16} 109^{2} 8167 x^{14}(\theta+1)(\theta+2)(\theta+3)(\theta+4)\)

Maple LaTex

Coefficients of the holomorphic solution: 1, 0, 252, 11106, 735660, ...
--> OEIS
Normalized instanton numbers (n₀=1): 343/26, 1358/13, 27836/13, 764852/13, 52338075/26, ... ; Common denominator:...

Discriminant

$-(661527z^5+290250z^4+47223z^3+3291z^2+71z-1)(23544z^3+7353z^2+759z+26)^2(9z+1)^3$

Local exponents

≈$-0.126194$ $-\frac{ 1}{ 9}$ ≈$-0.093057-0.009552I$ ≈$-0.093057+0.009552I$ $0$ $#ND+#NDI$ $\infty$
$0$ $0$ $0$ $0$ $0$ $0$ $1$
$1$ $0$ $1$ $1$ $0$ $1$ $2$
$3$ $0$ $3$ $3$ $0$ $1$ $3$
$4$ $0$ $4$ $4$ $0$ $2$ $4$

≈\(-0.126194\)	\(-\frac{ 1}{ 9}\)	≈\(-0.093057-0.009552I\)	≈\(-0.093057+0.009552I\)	\(0\)	\(#ND+#NDI\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(1\)
\(1\)	\(0\)	\(1\)	\(1\)	\(0\)	\(1\)	\(2\)
\(3\)	\(0\)	\(3\)	\(3\)	\(0\)	\(1\)	\(3\)
\(4\)	\(0\)	\(4\)	\(4\)	\(0\)	\(2\)	\(4\)

Note:

This is operator "14.6" from ...

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

5

New Number: 7.21 | AESZ: | Superseeker: 90 413926 | Hash: f2cdf32038c22a3da2f5752ad59eaa27

Degree: 7

$\theta^4-3^{2} x\left(27\theta^4+216\theta^3+234\theta^2+126\theta+28\right)-3^{6} x^{2}\left(75\theta^4-672\theta^3-2378\theta^2-2602\theta-1076\right)+3^{9} x^{3}\left(1843\theta^4+6360\theta^3-2836\theta^2-13692\theta-9828\right)-3^{14} x^{4}\left(373\theta^4+9344\theta^3+16396\theta^2+10260\theta+540\right)-3^{19} x^{5}\left(875\theta^4+152\theta^3-6794\theta^2-11462\theta-5400\right)+3^{26} x^{6}\left(71\theta^4+480\theta^3+1218\theta^2+1386\theta+600\right)+3^{33} x^{7}\left((\theta+2)^4\right)$

Maple LaTex

Coefficients of the holomorphic solution: 1, 252, 40419, 2460816, -1025424441, ...
--> OEIS
Normalized instanton numbers (n₀=1): 90, -4365, 413926, -38862153, 4502063682, ... ; Common denominator:...

Discriminant

$(1+27z)(243z+1)^2(59049z^2-378z+1)^2$

Local exponents

$-\frac{ 1}{ 27}$ $-\frac{ 1}{ 243}$ $0$ $\frac{ 7}{ 2187}-\frac{ 4}{ 2187}\sqrt{ 2}I$ $\frac{ 7}{ 2187}+\frac{ 4}{ 2187}\sqrt{ 2}I$ $\infty$
$0$ $0$ $0$ $0$ $0$ $2$
$1$ $1$ $0$ $-\frac{ 1}{ 2}$ $-\frac{ 1}{ 2}$ $2$
$1$ $3$ $0$ $1$ $1$ $2$
$2$ $4$ $0$ $\frac{ 3}{ 2}$ $\frac{ 3}{ 2}$ $2$

\(-\frac{ 1}{ 27}\)	\(-\frac{ 1}{ 243}\)	\(0\)	\(\frac{ 7}{ 2187}-\frac{ 4}{ 2187}\sqrt{ 2}I\)	\(\frac{ 7}{ 2187}+\frac{ 4}{ 2187}\sqrt{ 2}I\)	\(\infty\)
\(0\)	\(0\)	\(0\)	\(0\)	\(0\)	\(2\)
\(1\)	\(1\)	\(0\)	\(-\frac{ 1}{ 2}\)	\(-\frac{ 1}{ 2}\)	\(2\)
\(1\)	\(3\)	\(0\)	\(1\)	\(1\)	\(2\)
\(2\)	\(4\)	\(0\)	\(\frac{ 3}{ 2}\)	\(\frac{ 3}{ 2}\)	\(2\)

Note:

This is operator "7.21" 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:

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

\(12I\lambda+1\)	\(0+\frac{ 7}{ 4}I\)	\(-1\)	\(1\)
\(-\frac{ 1}{ 4}I\)	\(-\frac{ 9}{ 2}I\)	\(1\)	\(0\)
\(0\)	\(0+9I\)	\(0\)	\(0\)
\(-9I\)	\(0\)	\(0\)	\(0\)