### Summary

You searched for: sol=12

1-30  31-34

1

New Number: 2.20 |  AESZ: 133  |  Superseeker: 12 -3284/3  |  Hash: 4c9628f7dd48f4e9e6ec75303e557389

Degree: 2

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

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 324, 8400, 44100, ...
--> OEIS
Normalized instanton numbers (n0=1): 12, -42, -3284/3, -20538, -103776, ... ; Common denominator:...

#### Discriminant

$1-144z+6912z^2$

#### Local exponents

$0$$\frac{ 1}{ 96}-\frac{ 1}{ 288}\sqrt{ 3}I$$\frac{ 1}{ 96}+\frac{ 1}{ 288}\sqrt{ 3}I$$\infty$
$0$$0$$0$$\frac{ 1}{ 2}$
$0$$1$$1$$\frac{ 1}{ 2}$
$0$$1$$1$$\frac{ 3}{ 2}$
$0$$2$$2$$\frac{ 3}{ 2}$

#### Note:

Explicit solution not yet verified

2

New Number: 2.2 |  AESZ: 15  |  Superseeker: 21 15894  |  Hash: c8053e0e9c05ef468263fafd5e3fc764

Degree: 2

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

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 900, 94080, 11988900, ...
--> OEIS
Normalized instanton numbers (n0=1): 21, 480, 15894, 894075, 58703151, ... ; Common denominator:...

#### Discriminant

$-(27z+1)(216z-1)$

#### Local exponents

$-\frac{ 1}{ 27}$$0$$\frac{ 1}{ 216}$$\infty$
$0$$0$$0$$\frac{ 1}{ 3}$
$1$$0$$1$$\frac{ 2}{ 3}$
$1$$0$$1$$\frac{ 4}{ 3}$
$2$$0$$2$$\frac{ 5}{ 3}$

#### Note:

Hadamard product $B\ast a$.

A-Incarnation: diagonal of (3,3)-intersection in $P^2 \times P^2$

3

New Number: 2.5 |  AESZ: 25  |  Superseeker: 20 8220  |  Hash: 93279abcbeeade30c29508de7784e582

Degree: 2

$\theta^4-2^{2} x(2\theta+1)^2(11\theta^2+11\theta+3)-2^{4} x^{2}(2\theta+1)^2(2\theta+3)^2$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 684, 58800, 6129900, ...
--> OEIS
Normalized instanton numbers (n0=1): 20, 277, 8220, 352994, 18651536, ... ; Common denominator:...

#### Discriminant

$1-176z-256z^2$

#### Local exponents

$-\frac{ 11}{ 32}-\frac{ 5}{ 32}\sqrt{ 5}$$0$$-\frac{ 11}{ 32}+\frac{ 5}{ 32}\sqrt{ 5}$$\infty$
$0$$0$$0$$\frac{ 1}{ 2}$
$1$$0$$1$$\frac{ 1}{ 2}$
$1$$0$$1$$\frac{ 3}{ 2}$
$2$$0$$2$$\frac{ 3}{ 2}$

#### Note:

Hadamard product $A\ast b$

A-incarnation: X(1,2,2) in G(2,5)

4

New Number: 2.9 |  AESZ: 58  |  Superseeker: 16 11056/3  |  Hash: 1ca6d3d1c4514db0651efce420265f5a

Degree: 2

$\theta^4-2^{2} x(2\theta+1)^2(10\theta^2+10\theta+3)+2^{4} 3^{2} x^{2}(2\theta+1)^2(2\theta+3)^2$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 540, 37200, 3131100, ...
--> OEIS
Normalized instanton numbers (n0=1): 16, 142, 11056/3, 121470, 4971792, ... ; Common denominator:...

#### Discriminant

$(144z-1)(16z-1)$

#### Local exponents

$0$$\frac{ 1}{ 144}$$\frac{ 1}{ 16}$$\infty$
$0$$0$$0$$\frac{ 1}{ 2}$
$0$$1$$1$$\frac{ 1}{ 2}$
$0$$1$$1$$\frac{ 3}{ 2}$
$0$$2$$2$$\frac{ 3}{ 2}$

#### Note:

5

New Number: 3.22 |  AESZ: 392  |  Superseeker: 166 1016100  |  Hash: 5862be5cc4d3ec1686e6b9a6ec08f7e7

Degree: 3

$\theta^4-2 x\left(230\theta^4+496\theta^3+323\theta^2+75\theta+6\right)-2^{2} 3 x^{2}(6\theta+5)(1866\theta^3+5341\theta^2+4760\theta+1084)-2^{4} 3^{2} 13^{2} x^{3}(6\theta+5)(6\theta+11)(3\theta+1)(3\theta+7)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 5760, 1664544, 681014880, ...
--> OEIS
Normalized instanton numbers (n0=1): 166, 8076, 1016100, 189329096, 43879949258, ... ; Common denominator:...

#### Discriminant

$-(676z-1)(1+108z)^2$

#### Local exponents

$-\frac{ 1}{ 108}$$0$$\frac{ 1}{ 676}$$\infty$
$0$$0$$0$$\frac{ 1}{ 3}$
$\frac{ 1}{ 3}$$0$$1$$\frac{ 5}{ 6}$
$1$$0$$1$$\frac{ 11}{ 6}$
$\frac{ 4}{ 3}$$0$$2$$\frac{ 7}{ 3}$

#### Note:

This is operator "3.22" from ...

6

New Number: 3.28 |  AESZ: 410  |  Superseeker: 7 1057/3  |  Hash: accbbff67291992dfbc89e78f5a3c897

Degree: 3

$\theta^4-x\left(145\theta^4+242\theta^3+199\theta^2+78\theta+12\right)+2^{3} x^{2}(2\theta+1)(4\theta+3)(97\theta^2+182\theta+114)-2^{4} 3^{4} x^{3}(2\theta+1)(2\theta+3)(4\theta+3)(4\theta+7)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 336, 12880, 592200, ...
--> OEIS
Normalized instanton numbers (n0=1): 7, 22, 1057/3, 5460, 108241, ... ; Common denominator:...

#### Discriminant

$-(81z-1)(-1+32z)^2$

#### Local exponents

$0$$\frac{ 1}{ 81}$$\frac{ 1}{ 32}$$\infty$
$0$$0$$0$$\frac{ 1}{ 2}$
$0$$1$$\frac{ 3}{ 4}$$\frac{ 3}{ 4}$
$0$$1$$1$$\frac{ 3}{ 2}$
$0$$2$$\frac{ 7}{ 4}$$\frac{ 7}{ 4}$

#### Note:

This is operator "3.28" from ...

7

New Number: 4.58 |  AESZ: 282  |  Superseeker: 364/5 1264916  |  Hash: 582c9abe0a0b8176a2a06ec6c223bef4

Degree: 4

$5^{2} \theta^4-2^{2} 5 x\left(1348\theta^4+752\theta^3+521\theta^2+145\theta+15\right)+2^{4} 3^{4} x^{2}\left(5696\theta^4-1792\theta^3-7304\theta^2-3740\theta-585\right)-2^{10} 3^{8} x^{3}\left(20\theta^4-360\theta^3-289\theta^2-90\theta-10\right)-2^{12} 3^{13} x^{4}\left((2\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 3564, 1081200, 418200300, ...
--> OEIS
Normalized instanton numbers (n0=1): 364/5, 43384/5, 1264916, 1297643028/5, 323354425968/5, ... ; Common denominator:...

#### Discriminant

$-(62208z^2+560z-1)(-5+1296z)^2$

#### Local exponents

$-\frac{ 35}{ 7776}-\frac{ 13}{ 7776}\sqrt{ 13}$$0$$s_1$$s_2$$-\frac{ 35}{ 7776}+\frac{ 13}{ 7776}\sqrt{ 13}$$\frac{ 5}{ 1296}$$\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 hiding at infinity,corresponding to the Operator AESZ 283/4.59

8

New Number: 4.71 |  AESZ: 353  |  Superseeker: -4 -1580/9  |  Hash: 33845d8200fe810109063e352fbfc8b1

Degree: 4

$\theta^4-2^{2} x\left(52\theta^4+40\theta^3+37\theta^2+17\theta+3\right)+2^{4} x^{2}\left(960\theta^4+1536\theta^3+1512\theta^2+688\theta+123\right)-2^{8} x^{3}\left(1792\theta^4+4608\theta^3+5184\theta^2+2816\theta+597\right)+2^{14} x^{4}(4\theta+5)^2(4\theta+3)^2$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 324, 11856, 504900, ...
--> OEIS
Normalized instanton numbers (n0=1): -4, -24, -1580/9, -1580, -17120, ... ; Common denominator:...

#### Discriminant

$(16z-1)(64z-1)^3$

#### Local exponents

$0$$\frac{ 1}{ 64}$$\frac{ 1}{ 16}$$\infty$
$0$$0$$0$$\frac{ 3}{ 4}$
$0$$\frac{ 1}{ 2}$$1$$\frac{ 3}{ 4}$
$0$$\frac{ 3}{ 2}$$1$$\frac{ 5}{ 4}$
$0$$2$$2$$\frac{ 5}{ 4}$

#### Note:

Sporadic Operator, reducible to 3.33, so not a primary operator.

9

New Number: 4.72 |  AESZ: 361  |  Superseeker: 20 -119332/9  |  Hash: f55eaa640956f064f5230c04d8173d60

Degree: 4

$\theta^4-2^{2} x\left(80\theta^4+88\theta^3+67\theta^2+23\theta+3\right)+2^{4} 3 x^{2}\left(928\theta^4+2080\theta^3+2176\theta^2+972\theta+153\right)-2^{10} 3^{2} x^{3}\left(272\theta^4+648\theta^3+511\theta^2+162\theta+18\right)+2^{12} 3^{6} x^{4}\left((2\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 324, -6000, -2778300, ...
--> OEIS
Normalized instanton numbers (n0=1): 20, -139, -119332/9, -462222, -2113440, ... ; Common denominator:...

#### Discriminant

$(20736z^2-224z+1)(-1+48z)^2$

#### Local exponents

$0$$s_1$$s_2$$\frac{ 7}{ 1296}-\frac{ 1}{ 324}\sqrt{ 2}I$$\frac{ 7}{ 1296}+\frac{ 1}{ 324}\sqrt{ 2}I$$\frac{ 1}{ 48}$$\infty$
$0$$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$0$$1$$1$$1$$1$$1$$\frac{ 1}{ 2}$
$0$$1$$1$$1$$1$$3$$\frac{ 1}{ 2}$
$0$$2$$2$$2$$2$$4$$\frac{ 1}{ 2}$

#### Note:

Sporadic Operator. There is a second MUM-point hiding at infinity, corresponding to Operator AESZ 362/4.73

10

New Number: 5.35 |  AESZ: 218  |  Superseeker: 138/7 42984/7  |  Hash: a76111af659715caf2c4344eedd9d678

Degree: 5

$7^{2} \theta^4-2 3 7 x\left(192\theta^4+396\theta^3+303\theta^2+105\theta+14\right)+2^{2} 3 x^{2}\left(1188\theta^4+11736\theta^3+20431\theta^2+12152\theta+2436\right)+2^{2} 3^{3} x^{3}\left(532\theta^4+504\theta^3-3455\theta^2-3829\theta-1036\right)-2^{4} 3^{4} x^{4}(2\theta+1)(36\theta^3+306\theta^2+421\theta+156)-2^{6} 3^{4} x^{5}(2\theta+1)(3\theta+2)(3\theta+4)(2\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 612, 48000, 4580100, ...
--> OEIS
Normalized instanton numbers (n0=1): 138/7, 1506/7, 42984/7, 235596, 78950334/7, ... ; Common denominator:...

#### Discriminant

$-(1296z^3-864z^2+168z-1)(7+12z)^2$

#### Local exponents

$-\frac{ 7}{ 12}$$0$ ≈$0.006145$ ≈$0.330261-0.128447I$ ≈$0.330261+0.128447I$$\infty$
$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$1$$0$$1$$1$$1$$\frac{ 2}{ 3}$
$3$$0$$1$$1$$1$$\frac{ 4}{ 3}$
$4$$0$$2$$2$$2$$\frac{ 3}{ 2}$

#### Note:

This is operator "5.35" from ...

11

New Number: 5.36 |  AESZ: 219  |  Superseeker: 166/5 360988/15  |  Hash: b7068bb339f61ebd7c591b7be3fe5893

Degree: 5

$5^{2} \theta^4-2 5 x\left(464\theta^4+1036\theta^3+763\theta^2+245\theta+30\right)-2^{2} 3^{2} x^{2}\left(7064\theta^4+22472\theta^3+26699\theta^2+13200\theta+2340\right)-2^{4} 3^{4} x^{3}\left(3440\theta^4+13320\theta^3+18784\theta^2+10665\theta+2070\right)-2^{6} 3^{8} x^{4}(19\theta^2+59\theta+45)(2\theta+1)^2-2^{8} 3^{9} x^{5}(2\theta+1)^2(2\theta+3)^2$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 972, 109200, 14949900, ...
--> OEIS
Normalized instanton numbers (n0=1): 166/5, 638, 360988/15, 7222128/5, 524377242/5, ... ; Common denominator:...

#### Discriminant

$-(16z+1)(3888z^2+216z-1)(5+36z)^2$

#### Local exponents

$-\frac{ 5}{ 36}$$-\frac{ 1}{ 16}$$-\frac{ 1}{ 36}-\frac{ 1}{ 54}\sqrt{ 3}$$0$$-\frac{ 1}{ 36}+\frac{ 1}{ 54}\sqrt{ 3}$$\infty$
$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$1$$1$$1$$0$$1$$\frac{ 1}{ 2}$
$3$$1$$1$$0$$1$$\frac{ 3}{ 2}$
$4$$2$$2$$0$$2$$\frac{ 3}{ 2}$

#### Note:

This is operator "5.36" from ...

12

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}$

#### Note:

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

13

New Number: 5.40 |  AESZ: 226  |  Superseeker: 62/5 4060/3  |  Hash: 92f95cd33ac4bf18c2d05ce3040c5203

Degree: 5

$5^{2} \theta^4-2 5 x\left(328\theta^4+692\theta^3+551\theta^2+205\theta+30\right)+2^{2} 3 x^{2}\left(5352\theta^4+25416\theta^3+38387\theta^2+23020\theta+4860\right)-2^{4} 3^{3} x^{3}\left(352\theta^4+4520\theta^3+12108\theta^2+10205\theta+2630\right)-2^{6} 3^{3} x^{4}(2\theta+1)(586\theta^3+3039\theta^2+3947\theta+1527)-2^{8} 3^{4} x^{5}(2\theta+1)(6\theta+5)(6\theta+7)(2\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 396, 19920, 1241100, ...
--> OEIS
Normalized instanton numbers (n0=1): 62/5, 55, 4060/3, 28790, 861786, ... ; Common denominator:...

#### Discriminant

$-(16z-1)(108z-1)(12z-1)(5+12z)^2$

#### Local exponents

$-\frac{ 5}{ 12}$$0$$\frac{ 1}{ 108}$$\frac{ 1}{ 16}$$\frac{ 1}{ 12}$$\infty$
$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$1$$0$$1$$1$$1$$\frac{ 5}{ 6}$
$3$$0$$1$$1$$1$$\frac{ 7}{ 6}$
$4$$0$$2$$2$$2$$\frac{ 3}{ 2}$

#### Note:

This is operator "5.40" from ...

14

New Number: 5.44 |  AESZ: 240  |  Superseeker: 231/13 38037/13  |  Hash: 8f46cd6968b3b676e251a9d8635637fc

Degree: 5

$13^{2} \theta^4-13 x\left(1449\theta^4+4050\theta^3+3143\theta^2+1118\theta+156\right)-2^{4} x^{2}\left(22760\theta^4-27112\theta^3-121046\theta^2-82316\theta-17589\right)+2^{8} x^{3}\left(3824\theta^4+39936\theta^3-34292\theta^2-63492\theta-19539\right)-2^{16} 3 x^{4}(2\theta+1)(40\theta^3+684\theta^2+1013\theta+399)-2^{20} 3^{2} x^{5}(2\theta+1)(4\theta+3)(4\theta+5)(2\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 468, 28560, 2135700, ...
--> OEIS
Normalized instanton numbers (n0=1): 231/13, 826/13, 38037/13, 786076/13, 32662752/13, ... ; Common denominator:...

#### Discriminant

$-(128z-1)(128z^2-13z+1)(13+192z)^2$

#### Local exponents

$-\frac{ 13}{ 192}$$0$$\frac{ 1}{ 128}$$\frac{ 13}{ 256}-\frac{ 7}{ 256}\sqrt{ 7}I$$\frac{ 13}{ 256}+\frac{ 7}{ 256}\sqrt{ 7}I$$\infty$
$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$1$$0$$1$$1$$1$$\frac{ 3}{ 4}$
$3$$0$$1$$1$$1$$\frac{ 5}{ 4}$
$4$$0$$2$$2$$2$$\frac{ 3}{ 2}$

#### Note:

This is operator "5.44" from ...

15

New Number: 5.66 |  AESZ: 274  |  Superseeker: 49/5 6032/15  |  Hash: 729d44a3b7b561b49603f26a25d26069

Degree: 5

$5^{2} \theta^4-5 x\left(757\theta^4+1298\theta^3+1049\theta^2+400\theta+60\right)+2^{2} 3^{2} x^{2}\left(5456\theta^4+17498\theta^3+22121\theta^2+11940\theta+2340\right)-2^{2} 3^{4} x^{3}\left(15128\theta^4+68040\theta^3+112171\theta^2+73845\theta+16380\right)+2^{4} 3^{8} x^{4}(2\theta+1)(216\theta^3+864\theta^2+1015\theta+356)-2^{6} 3^{10} x^{5}(2\theta+1)(3\theta+2)(3\theta+4)(2\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 324, 12000, 548100, ...
--> OEIS
Normalized instanton numbers (n0=1): 49/5, -68/5, 6032/15, 36276/5, 350082/5, ... ; Common denominator:...

#### Discriminant

$-(81z-1)(1296z^2-56z+1)(-5+36z)^2$

#### Local exponents

$0$$\frac{ 1}{ 81}$$\frac{ 7}{ 324}-\frac{ 1}{ 81}\sqrt{ 2}I$$\frac{ 7}{ 324}+\frac{ 1}{ 81}\sqrt{ 2}I$$\frac{ 5}{ 36}$$\infty$
$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$0$$1$$1$$1$$1$$\frac{ 2}{ 3}$
$0$$1$$1$$1$$3$$\frac{ 4}{ 3}$
$0$$2$$2$$2$$4$$\frac{ 3}{ 2}$

#### Note:

This is operator "5.66" from ...

16

New Number: 5.75 |  AESZ: 298  |  Superseeker: 205/9 97622/9  |  Hash: e52d50673ec5c795512e2bc3e1017b12

Degree: 5

$3^{4} \theta^4-3^{2} x\left(1993\theta^4+3218\theta^3+2437\theta^2+828\theta+108\right)+2^{5} x^{2}\left(17486\theta^4+25184\theta^3+12239\theta^2+2790\theta+297\right)-2^{8} x^{3}\left(23620\theta^4+34776\theta^3+28905\theta^2+12447\theta+2106\right)+2^{15} x^{4}(2\theta+1)(340\theta^3+618\theta^2+455\theta+129)-2^{22} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 708, 63840, 6989220, ...
--> OEIS
Normalized instanton numbers (n0=1): 205/9, 3206/9, 97622/9, 496806, 254037095/9, ... ; Common denominator:...

#### Discriminant

$-(z-1)(1024z^2-192z+1)(-9+128z)^2$

#### Local exponents

$0$$\frac{ 3}{ 32}-\frac{ 1}{ 16}\sqrt{ 2}$$\frac{ 9}{ 128}$$\frac{ 3}{ 32}+\frac{ 1}{ 16}\sqrt{ 2}$$1$$\infty$
$0$$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:

This is operator "5.75" from ...

17

New Number: 5.76 |  AESZ: 306  |  Superseeker: 73/3 11119  |  Hash: d14307aa38b16c728ee31e5936937c44

Degree: 5

$3^{2} \theta^4-3 x\left(592\theta^4+1100\theta^3+829\theta^2+279\theta+36\right)+x^{2}\left(13801\theta^4+6652\theta^3-18041\theta^2-14904\theta-3312\right)-2 x^{3}\theta(8461\theta^3-29160\theta^2-28365\theta-7236)-2^{2} 3 7 x^{4}\left(513\theta^4+864\theta^3+487\theta^2+64\theta-16\right)-2^{3} 3 7^{2} x^{5}(\theta+1)^2(3\theta+2)(3\theta+4)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 732, 67080, 7456140, ...
--> OEIS
Normalized instanton numbers (n0=1): 73/3, 2131/6, 11119, 518671, 29749701, ... ; Common denominator:...

#### Discriminant

$-(z+1)(54z^2+189z-1)(-3+14z)^2$

#### Local exponents

$-\frac{ 7}{ 4}-\frac{ 11}{ 36}\sqrt{ 33}$$-1$$0$$-\frac{ 7}{ 4}+\frac{ 11}{ 36}\sqrt{ 33}$$\frac{ 3}{ 14}$$\infty$
$0$$0$$0$$0$$0$$\frac{ 2}{ 3}$
$1$$1$$0$$1$$1$$1$
$1$$1$$0$$1$$3$$1$
$2$$2$$0$$2$$4$$\frac{ 4}{ 3}$

#### Note:

This is operator "5.76" from ...

18

New Number: 11.1 |  AESZ:  |  Superseeker: -4 550/3  |  Hash: 9e36d74a520997fe52f0cbbfafae6aaf

Degree: 11

$\theta^4+x\left(6+38\theta+96\theta^2+116\theta^3+91\theta^4\right)+x^{2}\left(1218+5950\theta+11076\theta^2+9388\theta^3+3649\theta^4\right)+x^{3}\left(32814+148542\theta+258070\theta^2+203832\theta^3+63585\theta^4\right)+2 x^{4}\left(244543\theta^4+938432\theta^3+1417427\theta^2+933049\theta+226317\right)+2^{2} x^{5}\left(374407\theta^4+1908784\theta^3+3407293\theta^2+2501538\theta+653454\right)+2^{2} 3 x^{6}\left(130530\theta^4+686256\theta^3+1382165\theta^2+1159645\theta+333030\right)+2^{3} x^{7}\left(276464\theta^4-92912\theta^3-3194335\theta^2-3755703\theta-1224450\right)+2^{4} x^{8}\left(341712\theta^4+1614816\theta^3+1576879\theta^2+219863\theta-145632\right)-2^{5} x^{9}\left(29968\theta^4+412128\theta^3+489227\theta^2+156573\theta-3258\right)+2^{8} 3 x^{10}\left(6368\theta^4+13600\theta^3+11014\theta^2+4187\theta+681\right)-2^{11} 3^{2} x^{11}(4\theta+3)(\theta+1)^2(4\theta+5)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, -6, 54, 12, -26010, ...
--> OEIS
Normalized instanton numbers (n0=1): -4, -11, 550/3, -2965/2, -3316, ... ; Common denominator:...

#### Discriminant

$-(4z+1)(3z+1)(96z^3-1576z^2-62z-1)(1+11z-6z^2+16z^3)^2$

#### Local exponents

$-\frac{ 1}{ 3}$$-\frac{ 1}{ 4}$ ≈$-0.085955$ ≈$-0.019642-0.015722I$ ≈$-0.019642+0.015722I$$0$ ≈$0.230478-0.820976I$ ≈$0.230478+0.820976I$ ≈$16.455951$$\infty$
$0$$0$$0$$0$$0$$0$$0$$0$$0$$\frac{ 3}{ 4}$
$1$$1$$1$$1$$1$$0$$1$$1$$1$$1$
$1$$1$$3$$1$$1$$0$$3$$3$$1$$1$
$2$$2$$4$$2$$2$$0$$4$$4$$2$$\frac{ 5}{ 4}$

#### Note:

This is operator "11.1" from ...

19

New Number: 11.4 |  AESZ:  |  Superseeker: 116/5 29628/5  |  Hash: 4222cdacde3dbaf06ed32adadb70f0d6

Degree: 11

$5^{2} \theta^4-2^{2} 5 x\left(197\theta^4+418\theta^3+319\theta^2+110\theta+15\right)+2^{4} x^{2}\left(181\theta^4+5068\theta^3+10291\theta^2+6750\theta+1585\right)-2^{6} x^{3}\left(1727\theta^4-4758\theta^3-11365\theta^2-4560\theta-345\right)+2^{9} x^{4}\left(2351\theta^4+4552\theta^3-11125\theta^2-12552\theta-3833\right)-2^{12} x^{5}\left(527\theta^4+1448\theta^3+16\theta^2-1811\theta-887\right)+2^{15} x^{6}\left(493\theta^4-1527\theta^3-789\theta^2-363\theta-116\right)-2^{17} x^{7}\left(780\theta^4-282\theta^3+865\theta^2+1459\theta+563\right)+2^{20} x^{8}\left(151\theta^4-104\theta^3-291\theta^2-239\theta-65\right)-2^{22} x^{9}\left(23\theta^4+24\theta^3+85\theta^2+132\theta+55\right)+2^{25} x^{10}(\theta+1)(7\theta^3+31\theta^2+35\theta+12)-2^{28} x^{11}\left((\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 572, 42960, 3944556, ...
--> OEIS
Normalized instanton numbers (n0=1): 116/5, 1059/5, 29628/5, 2227181/10, 51562768/5, ... ; Common denominator:...

#### Discriminant

$-(-1+156z+160z^2+256z^3)(4z-1)^2(256z^3-128z^2-16z-5)^2$

#### Local exponents

≈$-0.315684-0.716756I$ ≈$-0.315684+0.716756I$ ≈$-0.072055-0.158527I$ ≈$-0.072055+0.158527I$$0$ ≈$0.006368$$\frac{ 1}{ 4}$ ≈$0.64411$$\infty$
$0$$0$$0$$0$$0$$0$$0$$0$$1$
$1$$1$$1$$1$$0$$1$$\frac{ 1}{ 2}$$1$$1$
$1$$1$$3$$3$$0$$1$$\frac{ 1}{ 2}$$3$$1$
$2$$2$$4$$4$$0$$2$$1$$4$$1$

#### Note:

This is operator "11.4" from ...

20

New Number: 12.4 |  AESZ:  |  Superseeker: 4 -228/5  |  Hash: c24070a1d4a449404cd7b46398fa6d6e

Degree: 12

$5^{2} \theta^4-2^{2} 5^{2} x\left(16\theta^4+32\theta^3+31\theta^2+15\theta+3\right)+2^{4} 5 x^{2}\left(736\theta^4+2368\theta^3+3848\theta^2+2960\theta+915\right)-2^{10} 5 x^{3}\left(304\theta^4+1176\theta^3+2337\theta^2+2313\theta+891\right)+2^{12} 3 x^{4}\left(2608\theta^4+10688\theta^3+21652\theta^2+23580\theta+9945\right)-2^{16} 3 x^{5}\left(2784\theta^4+11616\theta^3+21812\theta^2+22396\theta+9191\right)+2^{21} 3 x^{6}\left(1232\theta^4+5232\theta^3+9332\theta^2+7968\theta+2649\right)-2^{25} 3^{2} x^{7}\left(304\theta^4+1312\theta^3+2472\theta^2+1992\theta+559\right)+2^{30} 3 x^{8}\left(280\theta^4+1216\theta^3+2491\theta^2+2337\theta+827\right)-2^{32} x^{9}\left(1664\theta^4+7200\theta^3+13692\theta^2+11988\theta+3951\right)+2^{38} x^{10}\left(164\theta^4+832\theta^3+1751\theta^2+1731\theta+663\right)-2^{40} x^{11}\left(160\theta^4+928\theta^3+2072\theta^2+2072\theta+777\right)+2^{44} x^{12}\left((2\theta+3)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 108, 688, 3564, ...
--> OEIS
Normalized instanton numbers (n0=1): 4, -29/5, -228/5, 3724/5, -31856/5, ... ; Common denominator:...

#### Discriminant

$(16z-1)^2(256z^2-16z+1)^2(4096z^3-768z^2-5)^2$

#### Local exponents

≈$-0.013312-0.074322I$ ≈$-0.013312+0.074322I$$0$$\frac{ 1}{ 32}-\frac{ 1}{ 32}\sqrt{ 3}I$$\frac{ 1}{ 32}+\frac{ 1}{ 32}\sqrt{ 3}I$$\frac{ 1}{ 16}$ ≈$0.214124$$\infty$
$0$$0$$0$$0$$0$$0$$0$$\frac{ 3}{ 2}$
$1$$1$$0$$\frac{ 1}{ 2}$$\frac{ 1}{ 2}$$\frac{ 1}{ 2}$$1$$\frac{ 3}{ 2}$
$3$$3$$0$$\frac{ 1}{ 2}$$\frac{ 1}{ 2}$$\frac{ 1}{ 2}$$3$$\frac{ 3}{ 2}$
$4$$4$$0$$1$$1$$1$$4$$\frac{ 3}{ 2}$

#### Note:

This is operator "12.4" from ...

21

New Number: 13.10 |  AESZ:  |  Superseeker: 4 -628/9  |  Hash: 2a9fda379889eb2fd218bd01f2520f7a

Degree: 13

$\theta^4-2^{2} x\left(35\theta^4+38\theta^3+35\theta^2+16\theta+3\right)+2^{4} x^{2}\left(546\theta^4+1068\theta^3+1287\theta^2+790\theta+201\right)-2^{6} x^{3}\left(4928\theta^4+12888\theta^3+17829\theta^2+12673\theta+3693\right)+2^{8} x^{4}\left(28123\theta^4+88408\theta^3+131977\theta^2+98226\theta+29511\right)-2^{10} 3^{2} x^{5}\left(11315\theta^4+41094\theta^3+65088\theta^2+47691\theta+13532\right)+2^{13} 3^{2} x^{6}\left(11674\theta^4+48674\theta^3+79399\theta^2+52683\theta+11716\right)-2^{15} 3^{3} x^{7}\left(2063\theta^4+11102\theta^3+11184\theta^2-9217\theta-10762\right)-2^{17} 3^{4} x^{8}\left(3277\theta^4+16284\theta^3+42329\theta^2+57018\theta+27266\right)+2^{20} 3^{5} x^{9}\left(1124\theta^4+7114\theta^3+18121\theta^2+22265\theta+10018\right)+2^{24} 3^{6} x^{10}(\theta+1)(\theta^3-105\theta^2-277\theta-267)-2^{25} 3^{7} x^{11}(\theta+1)(\theta+2)(93\theta^2+441\theta+607)+2^{27} 3^{10} x^{12}(\theta+3)(\theta+2)(\theta+1)(\theta+6)+2^{30} 3^{10} x^{13}(\theta+1)(\theta+2)(\theta+3)(\theta+4)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 180, 2928, 47556, ...
--> OEIS
Normalized instanton numbers (n0=1): 4, 5, -628/9, -2823/4, 672, ... ; Common denominator:...

#### Discriminant

$(8z-1)(10368z^3-1728z^2+72z-1)(12z-1)^2(288z^2-24z+1)^2(4z+1)^3$

#### Local exponents

$-\frac{ 1}{ 4}$$0$ ≈$0.027033-0.011216I$ ≈$0.027033+0.011216I$$\frac{ 1}{ 24}-\frac{ 1}{ 24}I$$\frac{ 1}{ 24}+\frac{ 1}{ 24}I$$\frac{ 1}{ 12}$ ≈$0.112601$$\frac{ 1}{ 8}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$0$$0$$1$
$\frac{ 1}{ 2}$$0$$1$$1$$1$$1$$\frac{ 1}{ 2}$$1$$1$$2$
$\frac{ 3}{ 2}$$0$$1$$1$$3$$3$$\frac{ 1}{ 2}$$1$$1$$3$
$2$$0$$2$$2$$4$$4$$1$$2$$2$$4$

#### Note:

This is operator "13.10" from ...

22

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$

#### Note:

This is operator "15.4" from ...

23

New Number: 6.4 |  AESZ:  |  Superseeker: 370/19 140636/19  |  Hash: 0f3ddf420018e2870561a3e9fd2551cc

Degree: 6

$19^{2} \theta^4-19 x\left(4333\theta^4+6212\theta^3+4778\theta^2+1672\theta+228\right)+x^{2}\left(4307495\theta^4+7600484\theta^3+6216406\theta^2+2802424\theta+530556\right)-x^{3}\left(93729369\theta^4+213316800\theta^3+236037196\theta^2+125748612\theta+25260804\right)+2^{2} x^{4}\left(240813800\theta^4+778529200\theta^3+1041447759\theta^2+631802809\theta+138510993\right)-2^{2} 409 x^{5}(\theta+1)(2851324\theta^3+10035516\theta^2+11221241\theta+3481470)+2^{2} 3^{2} 19^{2} 409^{2} x^{6}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 588, 46200, 4446540, ...
--> OEIS
Normalized instanton numbers (n0=1): 370/19, 276, 140636/19, 5568700/19, 277119292/19, ... ; Common denominator:...

#### Discriminant

$(9z-1)(5776z^3-1920z^2+176z-1)(-19+409z)^2$

#### Local exponents

$0$ ≈$0.006077$$\frac{ 19}{ 409}$$\frac{ 1}{ 9}$ ≈$0.163166-0.043179I$ ≈$0.163166+0.043179I$$\infty$
$0$$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$0$$1$$1$$1$$1$$1$$1$
$0$$1$$3$$1$$1$$1$$2$
$0$$2$$4$$2$$2$$2$$\frac{ 5}{ 2}$

#### Note:

This is operator "6.4" from ...

24

New Number: 6.5 |  AESZ:  |  Superseeker: -11 -3422/3  |  Hash: 6a4aeb5833b7673c962d5598842d3f2c

Degree: 6

$\theta^4-x\left(12+64\theta+125\theta^2+122\theta^3+61\theta^4\right)-2^{3} x^{2}\left(193\theta^4+772\theta^3+1033\theta^2+522\theta+72\right)+2^{9} 3 x^{3}\left(146\theta^4+876\theta^3+1838\theta^2+1572\theta+405\right)-2^{12} 3^{2} x^{4}\left(204\theta^4+1632\theta^3+4449\theta^2+4740\theta+1400\right)+2^{16} 3^{3} x^{5}(16\theta^2+80\theta+35)(2\theta+5)^2-2^{19} 3^{4} x^{6}(2\theta+11)(2\theta+7)(2\theta+5)(2\theta+1)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 324, 5760, 215460, ...
--> OEIS
Normalized instanton numbers (n0=1): -11, 68, -3422/3, 30735, -1014993, ... ; Common denominator:...

#### Discriminant

$-(24z-1)(27648z^3-1728z^2+27z+1)(-1+32z)^2$

#### Local exponents

≈$-0.016119$$0$$\frac{ 1}{ 32}$ ≈$0.03931-0.026431I$ ≈$0.03931+0.026431I$$\frac{ 1}{ 24}$$\infty$
$0$$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$1$$0$$\frac{ 1}{ 2}$$1$$1$$1$$\frac{ 5}{ 2}$
$1$$0$$\frac{ 1}{ 2}$$1$$1$$1$$\frac{ 7}{ 2}$
$2$$0$$1$$2$$2$$2$$\frac{ 11}{ 2}$

#### Note:

This is operator "6.5" from ...

25

New Number: 6.6 |  AESZ:  |  Superseeker: 25 17452  |  Hash: e97e9b0e87960fe4cffbb22a5e935b4a

Degree: 6

$\theta^4-x\left(12+100\theta+305\theta^2+410\theta^3+205\theta^4\right)-2^{5} x^{2}\left(127\theta^4+508\theta^3+742\theta^2+468\theta+99\right)-2^{2} 3 x^{3}\left(2588\theta^4+15528\theta^3+32639\theta^2+28041\theta+7290\right)-2^{6} 3^{2} x^{4}\left(204\theta^4+1632\theta^3+4449\theta^2+4740\theta+1400\right)-2^{7} 3^{3} x^{5}(16\theta^2+80\theta+35)(2\theta+5)^2-2^{7} 3^{4} x^{6}(2\theta+11)(2\theta+7)(2\theta+5)(2\theta+1)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 972, 106200, 14027580, ...
--> OEIS
Normalized instanton numbers (n0=1): 25, 446, 17452, 958347, 65098152, ... ; Common denominator:...

#### Discriminant

$-(3z+1)(3456z^3+1728z^2+216z-1)(4z+1)^2$

#### Local exponents

$-\frac{ 1}{ 3}$ ≈$-0.252234-0.033647I$ ≈$-0.252234+0.033647I$$-\frac{ 1}{ 4}$$0$$\frac{ 1}{ 12}2^(\frac{ 1}{ 3})+\frac{ 1}{ 24}2^(\frac{ 2}{ 3})-\frac{ 1}{ 6}$$\infty$
$0$$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$1$$1$$1$$\frac{ 1}{ 2}$$0$$1$$\frac{ 5}{ 2}$
$1$$1$$1$$\frac{ 1}{ 2}$$0$$1$$\frac{ 7}{ 2}$
$2$$2$$2$$1$$0$$2$$\frac{ 11}{ 2}$

#### Note:

This is operator "6.6" from ...

26

New Number: 7.8 |  AESZ:  |  Superseeker: -1/3 -5/3  |  Hash: d5b8cfd5049e5d8670dac5bb5499d46a

Degree: 7

$3^{2} \theta^4-3 x\left(272\theta^4+340\theta^3+347\theta^2+177\theta+36\right)+x^{2}\left(31273\theta^4+76540\theta^3+103783\theta^2+71112\theta+19728\right)-2 x^{3}\left(328219\theta^4+1181160\theta^3+1977957\theta^2+1620036\theta+522288\right)+2^{2} x^{4}\left(2036999\theta^4+9602752\theta^3+19022113\theta^2+17726192\theta+6309408\right)-2^{3} 17 x^{5}(\theta+1)(439669\theta^3+2114103\theta^2+3708554\theta+2306280)+2^{6} 3^{3} 17^{2} x^{6}(\theta+1)(\theta+2)(481\theta^2+1875\theta+1962)-2^{10} 3^{4} 17^{3} x^{7}(\theta+1)(\theta+2)^2(\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 156, 2136, 30348, ...
--> OEIS
Normalized instanton numbers (n0=1): -1/3, 11/12, -5/3, 19/3, -29, ... ; Common denominator:...

#### Discriminant

$-(17z-1)(9z-1)(8z-1)(18z-1)(16z-1)(-3+34z)^2$

#### Local exponents

$0$$\frac{ 1}{ 18}$$\frac{ 1}{ 17}$$\frac{ 1}{ 16}$$\frac{ 3}{ 34}$$\frac{ 1}{ 9}$$\frac{ 1}{ 8}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$1$
$0$$1$$1$$1$$1$$1$$1$$2$
$0$$1$$1$$1$$3$$1$$1$$2$
$0$$2$$2$$2$$4$$2$$2$$3$

#### Note:

This is operator "7.8" from ...

27

New Number: 8.10 |  AESZ: 123  |  Superseeker: 12 1828/3  |  Hash: f0d76ab2b6b8808f4faa4ab8ecadff2c

Degree: 8

$\theta^4-2^{2} x(3\theta^2+3\theta+1)(10\theta^2+10\theta+3)+2^{4} x^{2}\left(209\theta^4+1052\theta^3+1471\theta^2+838\theta+183\right)+2^{7} 3^{2} x^{3}\left(30\theta^4-180\theta^3-551\theta^2-417\theta-111\right)-2^{10} 3^{2} x^{4}\left(227\theta^4+454\theta^3-550\theta^2-777\theta-261\right)+2^{12} 3^{4} x^{5}\left(30\theta^4+300\theta^3+169\theta^2-25\theta-35\right)+2^{14} 3^{4} x^{6}\left(209\theta^4-216\theta^3-431\theta^2-216\theta-27\right)-2^{17} 3^{6} x^{7}(3\theta^2+3\theta+1)(10\theta^2+10\theta+3)+2^{20} 3^{8} x^{8}\left((\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 300, 10416, 431964, ...
--> OEIS
Normalized instanton numbers (n0=1): 12, -47/2, 1828/3, -10813/4, 127948, ... ; Common denominator:...

#### Discriminant

$(36z-1)(8z-1)(72z-1)(4z-1)(-1+288z^2)^2$

#### Local exponents

$-\frac{ 1}{ 24}\sqrt{ 2}$$0$$\frac{ 1}{ 72}$$\frac{ 1}{ 36}$$\frac{ 1}{ 24}\sqrt{ 2}$$\frac{ 1}{ 8}$$\frac{ 1}{ 4}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$1$
$1$$0$$1$$1$$1$$1$$1$$1$
$3$$0$$1$$1$$3$$1$$1$$1$
$4$$0$$2$$2$$4$$2$$2$$1$

#### Note:

Hadamard product $c \ast d$. This operator has a second MUM-point at infinity with the same instanton numbers. It
can be reduced to an operator of degree 4 with a single
MUM-point defined over $\Q(\sqrt{?})$.

28

New Number: 8.13 |  AESZ: 163  |  Superseeker: 12 3020/3  |  Hash: e21fd830a9dca03305deb8363a26fcf2

Degree: 8

$\theta^4-2^{2} 3 x\left((3\theta^2+3\theta+1)^2\right)+2^{4} 3^{2} x^{2}\left(21\theta^4+156\theta^3+219\theta^2+126\theta+29\right)+2^{7} 3^{4} x^{3}(3\theta^2+3\theta+1)(3\theta^2-21\theta-35)-2^{10} 3^{5} x^{4}\left(27\theta^4+54\theta^3-114\theta^2-141\theta-49\right)+2^{12} 3^{7} x^{5}(3\theta^2+3\theta+1)(3\theta^2+27\theta-11)+2^{14} 3^{8} x^{6}\left(21\theta^4-72\theta^3-123\theta^2-72\theta-13\right)-2^{17} 3^{10} x^{7}\left((3\theta^2+3\theta+1)^2\right)+2^{20} 3^{12} x^{8}\left((\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 180, 2352, 6084, ...
--> OEIS
Normalized instanton numbers (n0=1): 12, -96, 3020/3, -71493/4, 319584, ... ; Common denominator:...

#### Discriminant

$(1728z^2-72z+1)(432z^2-36z+1)(-1+864z^2)^2$

#### Local exponents

$-\frac{ 1}{ 72}\sqrt{ 6}$$0$$\frac{ 1}{ 48}-\frac{ 1}{ 144}\sqrt{ 3}I$$\frac{ 1}{ 48}+\frac{ 1}{ 144}\sqrt{ 3}I$$\frac{ 1}{ 72}\sqrt{ 6}$$\frac{ 1}{ 24}-\frac{ 1}{ 72}\sqrt{ 3}I$$\frac{ 1}{ 24}+\frac{ 1}{ 72}\sqrt{ 3}I$$\infty$
$0$$0$$0$$0$$0$$0$$0$$1$
$1$$0$$1$$1$$1$$1$$1$$1$
$3$$0$$1$$1$$3$$1$$1$$1$
$4$$0$$2$$2$$4$$2$$2$$1$

#### Note:

Hadamard product $d \ast f$. This operator has a second MUM-point at infinity with the same instanton numbers. Itg can be reduced to an operator of degree 4 with a single MUM-point defined over $Q(\sqrt{?})$.

29

New Number: 8.35 |  AESZ: 326  |  Superseeker: 11/13 385/39  |  Hash: 946b91838924db64fe0ebdf0d473e621

Degree: 8

$13^{2} \theta^4-13 x\theta(56\theta^3+178\theta^2+115\theta+26)-x^{2}\left(28466\theta^4+109442\theta^3+165603\theta^2+117338\theta+32448\right)-x^{3}\left(233114\theta^4+1257906\theta^3+2622815\theta^2+2467842\theta+872352\right)-x^{4}\left(989585\theta^4+6852298\theta^3+17737939\theta^2+19969754\theta+8108448\right)-x^{5}(\theta+1)(2458967\theta^3+18007287\theta^2+44047582\theta+35386584)-3^{2} x^{6}(\theta+1)(\theta+2)(393163\theta^2+2539029\theta+4164444)-3^{3} 11 x^{7}(\theta+3)(\theta+2)(\theta+1)(8683\theta+34604)-3^{3} 11^{2} 13 17 x^{8}(\theta+1)(\theta+2)(\theta+3)(\theta+4)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 0, 12, 96, 1116, ...
--> OEIS
Normalized instanton numbers (n0=1): 11/13, 30/13, 385/39, 672/13, 4437/13, ... ; Common denominator:...

#### Discriminant

$-(3z+1)(13z^2+5z+1)(153z^3+75z^2+14z-1)(13+11z)^2$

#### Local exponents

$-\frac{ 13}{ 11}$$-\frac{ 1}{ 3}$ ≈$-0.272124-0.216493I$ ≈$-0.272124+0.216493I$$-\frac{ 5}{ 26}-\frac{ 3}{ 26}\sqrt{ 3}I$$-\frac{ 5}{ 26}+\frac{ 3}{ 26}\sqrt{ 3}I$$0$ ≈$0.054052$$\infty$
$0$$0$$0$$0$$0$$0$$0$$0$$1$
$1$$1$$1$$1$$1$$1$$0$$1$$2$
$3$$1$$1$$1$$1$$1$$0$$1$$3$
$4$$2$$2$$2$$2$$2$$0$$2$$4$

#### Note:

This opeerator is reducible to 6.25

30

New Number: 8.48 |  AESZ:  |  Superseeker: 359/13 393749/13  |  Hash: de7301c14448dbf584c01cc3722d0e58

Degree: 8

$13^{2} \theta^4-13 x\left(5249\theta^4+4930\theta^3+3687\theta^2+1222\theta+156\right)+2^{4} 3 x^{2}\left(175601\theta^4+188064\theta^3+90243\theta^2+19422\theta+1547\right)-2^{7} x^{3}\left(3336915\theta^4+3777024\theta^3+2377229\theta^2+746148\theta+94185\right)+2^{10} x^{4}\left(8591694\theta^4+11872968\theta^3+7381951\theta^2+2132674\theta+236280\right)-2^{12} x^{5}\left(15421829\theta^4+18326342\theta^3+7032841\theta^2+833608\theta-2718\right)+2^{16} 3^{2} x^{6}\left(334895\theta^4+615600\theta^3+867965\theta^2+590850\theta+138536\right)-2^{19} 3^{4} 7 x^{7}(\theta+1)(2\theta+1)(646\theta^2+1715\theta+1044)+2^{22} 3^{6} 7^{2} x^{8}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 852, 94800, 12860820, ...
--> OEIS
Normalized instanton numbers (n0=1): 359/13, 9162/13, 393749/13, 23364200/13, 1734245216/13, ... ; Common denominator:...

#### Discriminant

$(1-261z+6896z^2-6656z^3+36864z^4)(13-928z+4032z^2)^2$

#### Local exponents

$0$$\frac{ 29}{ 252}-\frac{ 1}{ 504}\sqrt{ 2545}$$\frac{ 29}{ 252}+\frac{ 1}{ 504}\sqrt{ 2545}$$#ND+#NDI$$\infty$
$0$$0$$0$$0$$\frac{ 1}{ 2}$
$0$$1$$1$$1$$1$
$0$$3$$3$$1$$1$
$0$$4$$4$$2$$\frac{ 3}{ 2}$

#### Note:

This is operator "8.48" from ...