Summary

You searched for: h3=4

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1

New Number: 4.43 |  AESZ: 225  |  Superseeker: 93984 25265152551072  |  Hash: 5993002ccf811247be9232b089dd8e3a  

Degree: 4

\(\theta^4+2^{4} x\left(22192\theta^4-17056\theta^3-9576\theta^2-1048\theta-49\right)+2^{20} x^{2}\left(33648\theta^4-44688\theta^3+16224\theta^2+1764\theta+17\right)+2^{34} 5 x^{3}\left(6512\theta^4-6144\theta^3-4440\theta^2-1536\theta-193\right)-2^{55} 5^{2} x^{4}\left((2\theta+1)^4\right)\)

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Coefficients of the holomorphic solution: 1, 784, 3226896, 20413907200, 157477179235600, ...
--> OEIS
Normalized instanton numbers (n0=1): 93984, -1084521600, 25265152551072, -787700706860008320, 28889437619654310485088, ... ; Common denominator:...

Discriminant

\(-(536870912z^2-27392z-1)(1+163840z)^2\)

Local exponents

≈\(-2.5e-05\)\(-\frac{ 1}{ 163840}\)\(0\)\(s_2\)\(s_1\) ≈\(7.6e-05\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(1\)\(3\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(2\)\(4\)\(0\)\(2\)\(2\)\(2\)\(\frac{ 1}{ 2}\)

Note:

Sporadic Operator. There is a second MUM-point
hiding at infinity, corresponding to Operator
AESZ 222/4.42

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2

New Number: 4.47 |  AESZ: 239  |  Superseeker: 1584 171534960  |  Hash: 8e610c3437d7f38e552038bc55399495  

Degree: 4

\(\theta^4+2^{4} 3 x\left(9\theta^4-198\theta^3-131\theta^2-32\theta-3\right)-2^{11} 3^{2} x^{2}\left(486\theta^4+1215\theta^3+81\theta^2-27\theta-5\right)-2^{16} 3^{5} x^{3}\left(891\theta^4+972\theta^3+675\theta^2+216\theta+25\right)-2^{23} 3^{8} x^{4}(3\theta+1)^2(3\theta+2)^2\)

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Coefficients of the holomorphic solution: 1, 144, 147600, 239904000, 479672701200, ...
--> OEIS
Normalized instanton numbers (n0=1): 1584, -17874, 171534960, 30012731550, 105934107802896, ... ; Common denominator:...

Discriminant

\(-(432z+1)(3456z-1)(1+1728z)^2\)

Local exponents

\(-\frac{ 1}{ 432}\)\(-\frac{ 1}{ 1728}\)\(0\)\(\frac{ 1}{ 3456}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 3}\)
\(1\)\(1\)\(0\)\(1\)\(\frac{ 1}{ 3}\)
\(1\)\(3\)\(0\)\(1\)\(\frac{ 2}{ 3}\)
\(2\)\(4\)\(0\)\(2\)\(\frac{ 2}{ 3}\)

Note:

Sporadic Operator.

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3

New Number: 4.48 |  AESZ: 241  |  Superseeker: 320 19748928  |  Hash: b4d16d8dd1eb7839630ecf8e8d242023  

Degree: 4

\(\theta^4-2^{4} x\left(152\theta^4+160\theta^3+110\theta^2+30\theta+3\right)+2^{10} 3 x^{2}\left(428\theta^4+176\theta^3-299\theta^2-170\theta-25\right)-2^{17} 3^{2} x^{3}\left(136\theta^4-216\theta^3-180\theta^2-51\theta-5\right)-2^{24} 3^{3} x^{4}(3\theta+1)(2\theta+1)^2(3\theta+2)\)

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Coefficients of the holomorphic solution: 1, 48, 26640, 21907200, 22048765200, ...
--> OEIS
Normalized instanton numbers (n0=1): 320, 61084, 19748928, 9428973876, 5618509433280, ... ; Common denominator:...

Discriminant

\(-(64z+1)(1728z-1)(-1+384z)^2\)

Local exponents

\(-\frac{ 1}{ 64}\)\(0\)\(\frac{ 1}{ 1728}\)\(\frac{ 1}{ 384}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 3}\)
\(1\)\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(1\)\(0\)\(1\)\(3\)\(\frac{ 1}{ 2}\)
\(2\)\(0\)\(2\)\(4\)\(\frac{ 2}{ 3}\)

Note:

Sporadic Operator.

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4

New Number: 4.51 |  AESZ:  |  Superseeker: 992 63721056  |  Hash: 1d45a05c9bcf007b5042b0f7a5672551  

Degree: 4

\(\theta^4-2^{4} x\left(112\theta^4+416\theta^3+280\theta^2+72\theta+7\right)-2^{12} x^{2}\left(656\theta^4+896\theta^3-216\theta^2-160\theta-23\right)-2^{23} x^{3}\left(96\theta^4+24\theta^3+12\theta^2+6\theta+1\right)-2^{30} x^{4}\left((2\theta+1)^4\right)\)

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Coefficients of the holomorphic solution: 1, 112, 93456, 124614400, 204621667600, ...
--> OEIS
Normalized instanton numbers (n0=1): 992, 98792, 63721056, 40943244128, 36122052633760, ... ; Common denominator:...

Discriminant

\(-(65536z^2+2816z-1)(1+512z)^2\)

Local exponents

\(-\frac{ 11}{ 512}-\frac{ 5}{ 512}\sqrt{ 5}\)\(-\frac{ 1}{ 512}\)\(0\)\(s_2\)\(s_1\)\(-\frac{ 11}{ 512}+\frac{ 5}{ 512}\sqrt{ 5}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(1\)\(3\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(2\)\(4\)\(0\)\(2\)\(2\)\(2\)\(\frac{ 1}{ 2}\)

Note:

Sporadic Operator. There is a second MUM-point
hiding at infinity, corresponding to Operator
AESZ 256/4.50

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5

New Number: 4.56 |  AESZ: 277  |  Superseeker: 4192 2124587232  |  Hash: 9d905a8d31566f4976cbeb2d3bf0624c  

Degree: 4

\(\theta^4+2^{4} x\left(576\theta^4-1152\theta^3-724\theta^2-148\theta-13\right)-2^{17} x^{2}\left(32\theta^4+992\theta^3-166\theta^2-57\theta-6\right)-2^{26} 3 x^{3}\left(832\theta^4+768\theta^3+556\theta^2+192\theta+25\right)-2^{40} 3^{2} x^{4}\left((2\theta+1)^4\right)\)

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Coefficients of the holomorphic solution: 1, 208, 254736, 490988800, 1163138813200, ...
--> OEIS
Normalized instanton numbers (n0=1): 4192, -1708008, 2124587232, -2777042329304, 4857272052090400, ... ; Common denominator:...

Discriminant

\(-(1024z+1)(4096z-1)(1+6144z)^2\)

Local exponents

\(-\frac{ 1}{ 1024}\)\(-\frac{ 1}{ 6144}\)\(0\)\(\frac{ 1}{ 4096}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(0\)\(1\)\(\frac{ 1}{ 2}\)
\(1\)\(3\)\(0\)\(1\)\(\frac{ 1}{ 2}\)
\(2\)\(4\)\(0\)\(2\)\(\frac{ 1}{ 2}\)

Note:

Sporadic Operator. There is a second MUM-point hiding at
infinity, corresponding to Operator 4.33, reducible to 3.35.

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6

New Number: 4.60 |  AESZ: 288  |  Superseeker: 3616 264403872  |  Hash: 3373ebdbe30d220b5562cfd77d4e8f96  

Degree: 4

\(\theta^4-2^{4} x\left(496\theta^4+1568\theta^3+1060\theta^2+276\theta+27\right)-2^{15} 3 x^{2}\left(32\theta^4-760\theta^3-1570\theta^2-651\theta-81\right)+2^{22} 3^{2} x^{3}\left(1616\theta^4+6912\theta^3+5092\theta^2+1416\theta+135\right)+2^{34} 3^{3} x^{4}(4\theta+1)(3\theta+1)(3\theta+2)(4\theta+3)\)

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Coefficients of the holomorphic solution: 1, 432, 982800, 3259872000, 12958462717200, ...
--> OEIS
Normalized instanton numbers (n0=1): 3616, 114144, 264403872, 424149521656, 710239010095456, ... ; Common denominator:...

Discriminant

\((6912z-1)(4096z-1)(1+1536z)^2\)

Local exponents

\(-\frac{ 1}{ 1536}\)\(0\)\(\frac{ 1}{ 6912}\)\(\frac{ 1}{ 4096}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 4}\)
\(1\)\(0\)\(1\)\(1\)\(\frac{ 1}{ 3}\)
\(3\)\(0\)\(1\)\(1\)\(\frac{ 2}{ 3}\)
\(4\)\(0\)\(2\)\(2\)\(\frac{ 3}{ 4}\)

Note:

Sporadic Operator.

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7

New Number: 4.64 |  AESZ: 295  |  Superseeker: -5408 -4296119968  |  Hash: e40629f953a095a2a764c68394321139  

Degree: 4

\(\theta^4-2^{4} x\left(816\theta^4-1440\theta^3-904\theta^2-184\theta-17\right)+2^{18} x^{2}\left(80\theta^4-592\theta^3+432\theta^2+164\theta+23\right)+2^{30} x^{3}\left(80\theta^4-384\theta^3-296\theta^2-96\theta-11\right)+2^{45} x^{4}\left((2\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -272, 93456, 194502400, -587215823600, ...
--> OEIS
Normalized instanton numbers (n0=1): -5408, -3839480, -4296119968, -6482749129792, -11816577914904160, ... ; Common denominator:...

Discriminant

\((8388608z^2+3328z+1)(-1+8192z)^2\)

Local exponents

\(-\frac{ 13}{ 65536}-\frac{ 7}{ 65536}\sqrt{ 7}I\)\(-\frac{ 13}{ 65536}+\frac{ 7}{ 65536}\sqrt{ 7}I\)\(0\)\(s_1\)\(s_2\)\(\frac{ 1}{ 8192}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(3\)\(\frac{ 1}{ 2}\)
\(2\)\(2\)\(0\)\(2\)\(2\)\(4\)\(\frac{ 1}{ 2}\)

Note:

Sporadic Operator. There is a second MUM-point hiding at
infinity, corresponding to Operator AESZ 296/4.65

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8

New Number: 4.66 |  AESZ: 300  |  Superseeker: -1616 -283183120  |  Hash: edc54887effd2ebcaa636dcc93baf0b7  

Degree: 4

\(\theta^4+2^{4} x\left(371\theta^4+862\theta^3+591\theta^2+160\theta+15\right)+2^{11} 5 x^{2}\left(224\theta^4+2069\theta^3+3277\theta^2+1363\theta+159\right)-2^{16} 5^{2} x^{3}\left(2089\theta^4+7500\theta^3+5533\theta^2+1500\theta+135\right)+2^{23} 5^{3} x^{4}(5\theta+1)(5\theta+2)(5\theta+3)(5\theta+4)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -240, 378000, -941740800, 2908743037200, ...
--> OEIS
Normalized instanton numbers (n0=1): -1616, 265534, -283183120, 351860487150, -525536710386800, ... ; Common denominator:...

Discriminant

\((6400000z^2+6576z+1)(-1+320z)^2\)

Local exponents

≈\(-0.000842\) ≈\(-0.000186\)\(0\)\(s_2\)\(s_1\)\(\frac{ 1}{ 320}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 5}\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(1\)\(\frac{ 2}{ 5}\)
\(1\)\(1\)\(0\)\(1\)\(1\)\(3\)\(\frac{ 3}{ 5}\)
\(2\)\(2\)\(0\)\(2\)\(2\)\(4\)\(\frac{ 4}{ 5}\)

Note:

Sporadic Operator.

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9

New Number: 1.10 |  AESZ: 10  |  Superseeker: 928 170869536  |  Hash: 51f8135aba94201bd0bbe9b2287a92d5  

Degree: 1

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

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Coefficients of the holomorphic solution: 1, 144, 176400, 341510400, 811620810000, ...
--> OEIS
Normalized instanton numbers (n0=1): 928, 245616, 170869536, 174999877936, 221984814405088, ... ; Common denominator:...

Discriminant

\(\)

No data for singularities

Note:

A-incarnation: X(4,4) in P^5(1,1,1,1,2,2)

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10

New Number: 1.14 |  AESZ: 14  |  Superseeker: 1248 683015008  |  Hash: 03af56f4ae0cea2c4b219620b08dc49b  

Degree: 1

\(\theta^4-2^{4} 3 x(6\theta+1)(2\theta+1)^2(6\theta+5)\)

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Coefficients of the holomorphic solution: 1, 240, 498960, 1633632000, 6558930378000, ...
--> OEIS
Normalized instanton numbers (n0=1): 1248, 597192, 683015008, 1149904141056, 2394928461766560, ... ; Common denominator:...

Discriminant

\(\)

No data for singularities

Note:

A-incarnation: X(2,6) in P^5(1,1,1,1,1,3)

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