### Summary

You searched for: sol=-12

1

New Number: 2.65 |  AESZ: 183  |  Superseeker: -4 -556/9  |  Hash: 04a3788c3f9ed53281ae824deb33d833

Degree: 2

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

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Coefficients of the holomorphic solution: 1, -12, 324, -11280, 447300, ...
--> OEIS
Normalized instanton numbers (n0=1): -4, 8, -556/9, 624, -8928, ... ; Common denominator:...

#### Discriminant

$(48z+1)(64z+1)$

#### Local exponents

$-\frac{ 1}{ 48}$$-\frac{ 1}{ 64}$$0$$\infty$
$0$$0$$0$$\frac{ 1}{ 2}$
$1$$1$$0$$\frac{ 3}{ 4}$
$1$$1$$0$$\frac{ 5}{ 4}$
$2$$2$$0$$\frac{ 3}{ 2}$

#### Note:

This is operator "2.65" from ...

2

New Number: 5.54 |  AESZ: 260  |  Superseeker: -188/5 -450516/5  |  Hash: 03ff8e2e94b897c3891e6981e7fb4ec9

Degree: 5

$5^{2} \theta^4+2^{2} 5 x\left(596\theta^4+544\theta^3+397\theta^2+125\theta+15\right)+2^{4} 3 x^{2}\left(30048\theta^4+14784\theta^3-13312\theta^2-10940\theta-2115\right)+2^{8} 3^{3} x^{3}\left(6368\theta^4-6720\theta^3-9052\theta^2-4080\theta-655\right)-2^{12} 3^{6} x^{4}(2\theta+1)^2(76\theta^2+196\theta+139)-2^{16} 3^{9} x^{5}(2\theta+1)^2(2\theta+3)^2$

Maple   LaTex

Coefficients of the holomorphic solution: 1, -12, 1260, -188400, 34353900, ...
--> OEIS
Normalized instanton numbers (n0=1): -188/5, 7693/5, -450516/5, 37785946/5, -790482672, ... ; Common denominator:...

#### Discriminant

$-(16z+1)(6912z^2-288z-1)(5+432z)^2$

#### Local exponents

$-\frac{ 1}{ 16}$$-\frac{ 5}{ 432}$$\frac{ 1}{ 48}-\frac{ 1}{ 72}\sqrt{ 3}$$0$$\frac{ 1}{ 48}+\frac{ 1}{ 72}\sqrt{ 3}$$\infty$
$0$$0$$0$$0$$0$$\frac{ 1}{ 2}$
$1$$1$$1$$0$$1$$\frac{ 1}{ 2}$
$1$$3$$1$$0$$1$$\frac{ 3}{ 2}$
$2$$4$$2$$0$$2$$\frac{ 3}{ 2}$

#### Note:

This is operator "5.54" from ...

3

New Number: 5.55 |  AESZ: 261  |  Superseeker: -76/5 -24836/5  |  Hash: adadb5e720011482371f48cfa73dab99

Degree: 5

$5^{2} \theta^4+2^{2} 5 x\left(292\theta^4+368\theta^3+289\theta^2+105\theta+15\right)+2^{4} x^{2}\left(24736\theta^4+43648\theta^3+38936\theta^2+18980\theta+3735\right)+2^{9} 3^{2} x^{3}\left(2512\theta^4+5760\theta^3+6328\theta^2+3330\theta+655\right)+2^{12} 3^{4} x^{4}(2\theta+1)(232\theta^3+588\theta^2+590\theta+207)+2^{18} 3^{6} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, -12, 492, -32880, 2743020, ...
--> OEIS
Normalized instanton numbers (n0=1): -76/5, 1103/5, -24836/5, 847456/5, -36542448/5, ... ; Common denominator:...

#### Discriminant

$(1+144z)(16z+1)^2(144z+5)^2$

#### Local exponents

$-\frac{ 1}{ 16}$$-\frac{ 5}{ 144}$$-\frac{ 1}{ 144}$$0$$\infty$
$0$$0$$0$$0$$\frac{ 1}{ 2}$
$\frac{ 1}{ 2}$$1$$1$$0$$1$
$\frac{ 1}{ 2}$$3$$1$$0$$1$
$1$$4$$2$$0$$\frac{ 3}{ 2}$

#### Note:

This is operator "5.55" from ...

4

New Number: 5.56 |  AESZ: 262  |  Superseeker: -28/5 -1268/5  |  Hash: 4899f97226a5ec3b1ded2994470e9fdc

Degree: 5

$5^{2} \theta^4+2^{2} 5 x\left(136\theta^4+224\theta^3+197\theta^2+85\theta+15\right)+2^{4} x^{2}\left(5584\theta^4+16192\theta^3+21924\theta^2+14800\theta+3955\right)+2^{11} x^{3}\left(608\theta^4+2280\theta^3+3642\theta^2+2745\theta+780\right)+2^{14} x^{4}\left(464\theta^4+1888\theta^3+2956\theta^2+2012\theta+501\right)+2^{24} x^{5}\left((\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, -12, 236, -6384, 217836, ...
--> OEIS
Normalized instanton numbers (n0=1): -28/5, 153/5, -1268/5, 18598/5, -320048/5, ... ; Common denominator:...

#### Discriminant

$(1+64z)(32z+5)^2(16z+1)^2$

#### Local exponents

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

#### Note:

There is a second MUM-point at infinity,
corresponding to the Operator AESZ 263/5.57

5

New Number: 12.5 |  AESZ:  |  Superseeker: 4 2252/9  |  Hash: bb257a283455fdd1fa17fef9649505e3

Degree: 12

$\theta^4+2^{2} x\left(43\theta^4+22\theta^3+25\theta^2+14\theta+3\right)+2^{4} x^{2}\left(753\theta^4+924\theta^3+1107\theta^2+622\theta+141\right)+2^{7} x^{3}\left(3377\theta^4+7218\theta^3+9261\theta^2+5764\theta+1455\right)+2^{10} x^{4}\left(7570\theta^4+24718\theta^3+34375\theta^2+21933\theta+5310\right)+2^{12} 3^{2} x^{5}\left(901\theta^4+5118\theta^3+5777\theta^2-84\theta-1829\right)-2^{14} 3^{2} x^{6}\left(7783\theta^4+33872\theta^3+83851\theta^2+107556\theta+49489\right)-2^{17} 3^{3} x^{7}\left(4895\theta^4+28154\theta^3+69267\theta^2+83564\theta+36929\right)-2^{20} 3^{4} x^{8}\left(44\theta^4+528\theta^3+247\theta^2+240\theta+274\right)+2^{23} 3^{5} x^{9}\left(664\theta^4+4760\theta^3+13781\theta^2+17353\theta+7679\right)+2^{26} 3^{6} x^{10}(\theta+1)(109\theta^3+651\theta^2+1373\theta+933)-2^{29} 3^{7} x^{11}(\theta+1)(\theta+2)(27\theta^2+153\theta+199)-2^{33} 3^{9} x^{12}(\theta+1)(\theta+2)^2(\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, -12, 180, -2736, 42948, ...
--> OEIS
Normalized instanton numbers (n0=1): 4, -31, 2252/9, -11109/4, 33312, ... ; Common denominator:...

#### Discriminant

$-(16z+1)(432z^2+36z+1)(24z+1)^2(288z^2+48z+1)^2(8z-1)^3$

#### Local exponents

$-\frac{ 1}{ 12}-\frac{ 1}{ 24}\sqrt{ 2}$$-\frac{ 1}{ 16}$$-\frac{ 1}{ 24}-\frac{ 1}{ 72}\sqrt{ 3}I$$-\frac{ 1}{ 24}$$-\frac{ 1}{ 24}+\frac{ 1}{ 72}\sqrt{ 3}I$$-\frac{ 1}{ 12}+\frac{ 1}{ 24}\sqrt{ 2}$$0$$\frac{ 1}{ 8}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$0$$1$
$1$$1$$1$$\frac{ 1}{ 2}$$1$$1$$0$$\frac{ 1}{ 2}$$2$
$3$$1$$1$$\frac{ 1}{ 2}$$1$$3$$0$$\frac{ 3}{ 2}$$2$
$4$$2$$2$$1$$2$$4$$0$$2$$3$

#### Note:

This is operator "12.5" from ...

6

New Number: 15.2 |  AESZ:  |  Superseeker: 2 38  |  Hash: 76d5c5e186c39f14d8f32dfa0f13e22a

Degree: 15

$\theta^4+2 x\left(73\theta^4+40\theta^3+47\theta^2+27\theta+6\right)+2^{2} x^{2}\left(2349\theta^4+2724\theta^3+3516\theta^2+2273\theta+614\right)+2^{3} x^{3}\left(44091\theta^4+80304\theta^3+115043\theta^2+84574\theta+26356\right)+2^{5} x^{4}\left(269591\theta^4+678346\theta^3+1084179\theta^2+893856\theta+309842\right)+2^{8} x^{5}\left(568775\theta^4+1835004\theta^3+3266314\theta^2+2971734\theta+1120498\right)+2^{11} x^{6}\left(856369\theta^4+3369012\theta^3+6631886\theta^2+6564309\theta+2651780\right)+2^{11} x^{7}\left(7508036\theta^4+34719008\theta^3+74840604\theta^2+79593816\theta+34039943\right)+2^{13} x^{8}\left(12098492\theta^4+63919352\theta^3+149239952\theta^2+168641212\theta+75569097\right)+2^{16} x^{9}\left(7179524\theta^4+42349744\theta^3+105902696\theta^2+125838704\theta+58525593\right)+2^{19} x^{10}\left(3117952\theta^4+20136896\theta^3+53326176\theta^2+65967996\theta+31556287\right)+2^{21} x^{11}\left(1949840\theta^4+13550976\theta^3+37571920\theta^2+47915544\theta+23371681\right)+2^{23} x^{12}\left(851424\theta^4+6266688\theta^3+17985676\theta^2+23424156\theta+11560933\right)+2^{26} x^{13}\left(122784\theta^4+942720\theta^3+2770088\theta^2+3654240\theta+1815239\right)+2^{31} 5 x^{14}\left(524\theta^4+4136\theta^3+12323\theta^2+16373\theta+8173\right)+2^{36} 5^{2} x^{15}\left((\theta+2)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, -12, 136, -1632, 21296, ...
--> OEIS
Normalized instanton numbers (n0=1): 2, -29/4, 38, -2077/8, 2034, ... ; Common denominator:...

#### Discriminant

$(4z+1)(64z^2+24z+1)^2(160z^2+32z+1)^2(2z+1)^3(8z+1)^3$

#### Local exponents

$-\frac{ 1}{ 2}$$-\frac{ 3}{ 16}-\frac{ 1}{ 16}\sqrt{ 5}$$-\frac{ 1}{ 4}$$-\frac{ 1}{ 10}-\frac{ 1}{ 40}\sqrt{ 6}$$-\frac{ 1}{ 8}$$-\frac{ 3}{ 16}+\frac{ 1}{ 16}\sqrt{ 5}$$-\frac{ 1}{ 10}+\frac{ 1}{ 40}\sqrt{ 6}$$0$$\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 "15.2" from ...

7

New Number: 8.84 |  AESZ:  |  Superseeker: 1/5 224/5  |  Hash: 258fab6f0a4f132fe597fc6f30e54eea

Degree: 8

$5^{2} \theta^4+5 x\theta^2(-1-2\theta+107\theta^2)+2^{2} x^{2}\left(2174\theta^4+5942\theta^3+8569\theta^2+5200\theta+1200\right)+2^{2} 3^{2} x^{3}\left(308\theta^4-4248\theta^3-17051\theta^2-16785\theta-5280\right)-2^{4} 3^{2} x^{4}\left(7060\theta^4+39500\theta^3+69820\theta^2+52851\theta+14688\right)-2^{6} 3^{4} x^{5}\left(881\theta^4+3974\theta^3+8648\theta^2+7983\theta+2581\right)+2^{7} 3^{4} x^{6}\left(1192\theta^4+2376\theta^3-1132\theta^2-4185\theta-1926\right)+2^{8} 3^{6} x^{7}\left(68\theta^4+568\theta^3+1095\theta^2+811\theta+210\right)-2^{12} 3^{8} x^{8}\left((\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 0, -12, 144, 324, ...
--> OEIS
Normalized instanton numbers (n0=1): 1/5, -6, 224/5, -448/5, -4334/5, ... ; Common denominator:...

#### Discriminant

$-(9z-1)(576z^3+368z^2+16z+1)(-5-36z+72z^2)^2$

#### Local exponents

≈$-0.597246$$\frac{ 1}{ 4}-\frac{ 1}{ 12}\sqrt{ 19}$ ≈$-0.020821-0.049733I$ ≈$-0.020821+0.049733I$$0$$\frac{ 1}{ 9}$$\frac{ 1}{ 4}+\frac{ 1}{ 12}\sqrt{ 19}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$1$
$1$$1$$1$$1$$0$$1$$1$$1$
$1$$3$$1$$1$$0$$1$$3$$1$
$2$$4$$2$$2$$0$$2$$4$$1$

#### Note:

This is operator "8.84" from ...