Summary

You searched for: Spectrum0=1/2,1/2,1/2,1/2

Your search produced 22 matches

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1

New Number: 4.33 |  AESZ: 55  |  Superseeker: 76/3 144196/3  |  Hash: 7e88cd5b7dc1c51022b66ac6f009218f  

Degree: 4

\(3^{2} \theta^4-2^{2} 3 x\left(208\theta^4+224\theta^3+163\theta^2+51\theta+6\right)+2^{9} x^{2}\left(32\theta^4-928\theta^3-1606\theta^2-837\theta-141\right)+2^{16} x^{3}\left(144\theta^4+576\theta^3+467\theta^2+144\theta+15\right)-2^{24} x^{4}\left((2\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 8, 936, 108800, 16748200, ...
--> OEIS
Normalized instanton numbers (n0=1): 76/3, 3476/3, 144196/3, 3563196, 309069600, ... ; Common denominator:...

Discriminant

\(-(64z+1)(256z-1)(-3+128z)^2\)

Local exponents

\(0\)\(\frac{ 3}{ 128}\)\(-\frac{ 1}{ 64}\)\(\infty\)\(\frac{ 1}{ 256}\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)\(0\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)
\(0\)\(3\)\(1\)\(\frac{ 1}{ 2}\)\(1\)
\(0\)\(4\)\(2\)\(\frac{ 1}{ 2}\)\(2\)

Note:

Sporadic operator. There is a second MUM-point
hiding at infinity, corresponding to Operator 4.56

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2

New Number: 4.34 |  AESZ: 99  |  Superseeker: 647/13 942613/13  |  Hash: f6c6b846edc829f336d8e4ae1dcb5618  

Degree: 4

\(13^{2} \theta^4-13 x\left(4569\theta^4+9042\theta^3+6679\theta^2+2158\theta+260\right)+2^{4} x^{2}\left(6386\theta^4-1774\theta^3-17898\theta^2-11596\theta-2119\right)+2^{8} x^{3}\left(67\theta^4+1248\theta^3+1091\theta^2+312\theta+26\right)-2^{12} x^{4}\left((2\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 20, 2196, 369200, 75562900, ...
--> OEIS
Normalized instanton numbers (n0=1): 647/13, 16166/13, 942613/13, 80218296/13, 8418215008/13, ... ; Common denominator:...

Discriminant

\(-(256z^2+349z-1)(-13+16z)^2\)

Local exponents

\(0\)\(-\frac{ 349}{ 512}-\frac{ 85}{ 512}\sqrt{ 17}\)\(-\frac{ 349}{ 512}+\frac{ 85}{ 512}\sqrt{ 17}\)\(\infty\)\(\frac{ 13}{ 16}\)\(s_1\)\(s_2\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)\(0\)\(0\)\(0\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(3\)\(1\)\(1\)
\(0\)\(2\)\(2\)\(\frac{ 1}{ 2}\)\(4\)\(2\)\(2\)

Note:

Sporadic Operator.
There is a second MUM point hidden at infinity. That is operator AESZ 207/4.38
A-Incarnation: $5 \times 5$-Pfaffian in P^5

A-Incarnation: 5 \times 5 Pfaffian in P^5

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3

New Number: 4.38 |  AESZ: 207  |  Superseeker: -70944 -3707752060576  |  Hash: eadc0882a9bf59840ef2b4a602f586e8  

Degree: 4

\(\theta^4-2^{4} x\left(1072\theta^4-17824\theta^3-10888\theta^2-1976\theta-145\right)-2^{17} x^{2}\left(51088\theta^4+116368\theta^3-45264\theta^2-14228\theta-1397\right)+2^{28} 13 x^{3}\left(73104\theta^4+1536\theta^3-488\theta^2+384\theta+97\right)-2^{44} 13^{2} x^{4}\left((2\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -2320, 57601296, -2373661139200, 121665506430000400, ...
--> OEIS
Normalized instanton numbers (n0=1): -70944, 107317768, -3707752060576, 66327758316665792, -1970671594871618215520, ... ; Common denominator:...

Discriminant

\(-(16777216z^2-89344z-1)(-1+53248z)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 53248}\)\(\frac{ 349}{ 131072}+\frac{ 85}{ 131072}\sqrt{ 17}\)\(\infty\)\(\frac{ 349}{ 131072}-\frac{ 85}{ 131072}\sqrt{ 17}\)\(s_2\)\(s_1\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)\(0\)\(0\)\(0\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)
\(0\)\(3\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)
\(0\)\(4\)\(2\)\(\frac{ 1}{ 2}\)\(2\)\(2\)\(2\)

Note:

Sporadic Operator.
There is a further MUM point hidden at infinity.
That operator is AESZ 99/4.34

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4

New Number: 4.39 |  AESZ: 210  |  Superseeker: -444/5 -1501908/5  |  Hash: 155d0198a5b26de08a0c2caf680f0786  

Degree: 4

\(5^{2} \theta^4+2^{2} 5 x\left(688\theta^4+1352\theta^3+981\theta^2+305\theta+35\right)+2^{4} x^{2}\left(5856\theta^4+7008\theta^3+96\theta^2-1260\theta-265\right)+2^{10} x^{3}\left(176\theta^4+120\theta^3+69\theta^2+30\theta+5\right)+2^{12} x^{4}\left((2\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -28, 4716, -1226800, 389349100, ...
--> OEIS
Normalized instanton numbers (n0=1): -444/5, 16653/5, -1501908/5, 199965534/5, -6573697776, ... ; Common denominator:...

Discriminant

\((256z^2+544z+1)(5+16z)^2\)

Local exponents

\(0\)\(-\frac{ 17}{ 16}-\frac{ 3}{ 4}\sqrt{ 2}\)\(\infty\)\(-\frac{ 17}{ 16}+\frac{ 3}{ 4}\sqrt{ 2}\)\(-\frac{ 5}{ 16}\)\(s_1\)\(s_2\)
\(0\)\(0\)\(\frac{ 1}{ 2}\)\(0\)\(0\)\(0\)\(0\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)\(1\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(3\)\(1\)\(1\)
\(0\)\(2\)\(\frac{ 1}{ 2}\)\(2\)\(4\)\(2\)\(2\)

Note:

Sporadic operator. There is a second MUM point hidden at infinity; Operator AESZ 211/4.40

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5

New Number: 4.40 |  AESZ: 211  |  Superseeker: -2400 -2956977632  |  Hash: c923c78e33e72a3a2b294bf3f2749298  

Degree: 4

\(\theta^4+2^{4} x\left(704\theta^4+928\theta^3+612\theta^2+148\theta+13\right)+2^{12} x^{2}\left(5856\theta^4+4704\theta^3-1632\theta^2-972\theta-121\right)+2^{20} 5 x^{3}\left(2752\theta^4+96\theta^3-60\theta^2+24\theta+7\right)+2^{28} 5^{2} x^{4}\left((2\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -208, 531216, -2168300800, 10900554288400, ...
--> OEIS
Normalized instanton numbers (n0=1): -2400, 1830480, -2956977632, 7117422755016, -21319886408804640, ... ; Common denominator:...

Discriminant

\((65536z^2+8704z+1)(1+1280z)^2\)

Local exponents

\(0\)\(-\frac{ 17}{ 256}+\frac{ 3}{ 64}\sqrt{ 2}\)\(\infty\)\(-\frac{ 17}{ 256}-\frac{ 3}{ 64}\sqrt{ 2}\)\(-\frac{ 1}{ 1280}\)\(s_1\)\(s_2\)
\(0\)\(0\)\(\frac{ 1}{ 2}\)\(0\)\(0\)\(0\)\(0\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)\(1\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(3\)\(1\)\(1\)
\(0\)\(2\)\(\frac{ 1}{ 2}\)\(2\)\(4\)\(2\)\(2\)

Note:

Sporadic Operator. There is a second MUM point hidden at infinity. That corresponds to Operator AESZ210/

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6

New Number: 4.42 |  AESZ: 222  |  Superseeker: 69/5 29081/5  |  Hash: aad7a72e711c9c463396d319e0bf7603  

Degree: 4

\(5^{2} \theta^4-5 x\left(407\theta^4+1198\theta^3+909\theta^2+310\theta+40\right)-2^{7} x^{2}\left(2103\theta^4+6999\theta^3+8358\theta^2+4050\theta+680\right)-2^{12} x^{3}\left(1387\theta^4+3840\theta^3+3081\theta^2+960\theta+100\right)-2^{21} x^{4}\left((2\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 8, 504, 36800, 3518200, ...
--> OEIS
Normalized instanton numbers (n0=1): 69/5, 1383/4, 29081/5, 346080, 72023607/5, ... ; Common denominator:...

Discriminant

\(-(8192z^2+107z-1)(5+64z)^2\)

Local exponents

\(0\)\(-\frac{ 107}{ 16384}-\frac{ 51}{ 16384}\sqrt{ 17}\)\(-\frac{ 107}{ 16384}+\frac{ 51}{ 16384}\sqrt{ 17}\)\(\infty\)\(-\frac{ 5}{ 64}\)\(s_1\)\(s_2\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)\(0\)\(0\)\(0\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(3\)\(1\)\(1\)
\(0\)\(2\)\(2\)\(\frac{ 1}{ 2}\)\(4\)\(2\)\(2\)

Note:

Sporadic Operator. There is a second MUM-point hiding at infinity, corresponding to Operator AESZ225/4.43

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7

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

Degree: 4

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

Maple   LaTex

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

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

Note:

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

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8

New Number: 4.50 |  AESZ: 256  |  Superseeker: -128 -800384  |  Hash: 05e172cfdecc836685981a2b01b75d1d  

Degree: 4

\(\theta^4+2^{5} x\left(24\theta^4+42\theta^3+30\theta^2+9\theta+1\right)+2^{8} x^{2}\left(164\theta^4+104\theta^3-144\theta^2-100\theta-17\right)+2^{14} x^{3}\left(28\theta^4-48\theta^3-44\theta^2-12\theta-1\right)-2^{18} x^{4}\left((2\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -32, 7056, -2393600, 991152400, ...
--> OEIS
Normalized instanton numbers (n0=1): -128, 6884, -800384, 143245314, -31691939200, ... ; Common denominator:...

Discriminant

\(-(4096z^2-704z-1)(1+32z)^2\)

Local exponents

\(0\)\(\frac{ 11}{ 128}+\frac{ 5}{ 128}\sqrt{ 5}\)\(\frac{ 11}{ 128}-\frac{ 5}{ 128}\sqrt{ 5}\)\(-\frac{ 1}{ 32}\)\(\infty\)\(s_1\)\(s_2\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)\(0\)\(0\)
\(0\)\(1\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)
\(0\)\(1\)\(1\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(1\)
\(0\)\(2\)\(2\)\(4\)\(\frac{ 1}{ 2}\)\(2\)\(2\)

Note:

Sporadic Operator. There is a second MUM-point hiding at infinity, corresponding to Operator AESZ 257/4.51
B-Incarnation:
Fibre product 4*11-- x 25311,
Double octic; D.O.257

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9

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

Degree: 4

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

Maple   LaTex

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

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

Note:

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

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10

New Number: 4.52 |  AESZ: 258  |  Superseeker: 480 4215904  |  Hash: bfb9f01124fd9980817cbf1b50f789c3  

Degree: 4

\(\theta^4-2^{4} x\left(16\theta^4+224\theta^3+156\theta^2+44\theta+5\right)-2^{14} x^{2}\left(48\theta^4+48\theta^3-120\theta^2-66\theta-11\right)-2^{22} x^{3}\left(16\theta^4-192\theta^3-156\theta^2-48\theta-5\right)+2^{32} x^{4}\left((2\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 80, 24336, 11398400, 6632189200, ...
--> OEIS
Normalized instanton numbers (n0=1): 480, -16536, 4215904, -242723592, 151800032928, ... ; Common denominator:...

Discriminant

\((1024z-1)(256z-1)(1+512z)^2\)

Local exponents

\(0\)\(-\frac{ 1}{ 512}\)\(\infty\)\(\frac{ 1}{ 256}\)\(\frac{ 1}{ 1024}\)
\(0\)\(0\)\(\frac{ 1}{ 2}\)\(0\)\(0\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)
\(0\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(1\)
\(0\)\(4\)\(\frac{ 1}{ 2}\)\(2\)\(2\)

Note:

Sporadic Operator.

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11

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

Degree: 4

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

Maple   LaTex

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

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

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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12

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

\(0\)\(-\frac{ 35}{ 7776}+\frac{ 13}{ 7776}\sqrt{ 13}\)\(-\frac{ 35}{ 7776}-\frac{ 13}{ 7776}\sqrt{ 13}\)\(\infty\)\(\frac{ 5}{ 1296}\)\(s_1\)\(s_2\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)\(0\)\(0\)\(0\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(3\)\(1\)\(1\)
\(0\)\(2\)\(2\)\(\frac{ 1}{ 2}\)\(4\)\(2\)\(2\)

Note:

Sporadic Operator. There is a second MUM-point hiding at infinity,corresponding to the Operator AESZ 283/4.59

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13

New Number: 4.59 |  AESZ: 283  |  Superseeker: -188 -807540  |  Hash: 987222cb05e8a3d02c76c47abefbb9f4  

Degree: 4

\(\theta^4+2^{2} x\left(20\theta^4+400\theta^3+281\theta^2+81\theta+9\right)-2^{4} 3 x^{2}\left(5696\theta^4+13184\theta^3+3928\theta^2+628\theta+39\right)+2^{10} 3^{2} 5 x^{3}\left(1348\theta^4+1944\theta^3+1415\theta^2+486\theta+63\right)-2^{12} 3^{7} 5^{2} x^{4}\left((2\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -36, 7236, -2257200, 860876100, ...
--> OEIS
Normalized instanton numbers (n0=1): -188, 831, -807540, 39235244, -18812436256, ... ; Common denominator:...

Discriminant

\(-(62208z^2-560z-1)(-1+240z)^2\)

Local exponents

\(0\)\(\frac{ 35}{ 7776}+\frac{ 13}{ 7776}\sqrt{ 13}\)\(\frac{ 35}{ 7776}-\frac{ 13}{ 7776}\sqrt{ 13}\)\(\infty\)\(\frac{ 1}{ 240}\)\(s_1\)\(s_2\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)\(0\)\(0\)\(0\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(3\)\(1\)\(1\)
\(0\)\(2\)\(2\)\(\frac{ 1}{ 2}\)\(4\)\(2\)\(2\)

Note:

Sporadic Operator. There is a second MUM-point hiding at
infinity, corresponding to AESZ 282/4.58

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14

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

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

Note:

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

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15

New Number: 4.65 |  AESZ:  |  Superseeker: 48 -9104  |  Hash: 5ec2790b5eda514313634b7aeb0a295c  

Degree: 4

\(\theta^4-2^{4} x\left(5\theta^4+34\theta^3+25\theta^2+8\theta+1\right)+2^{11} x^{2}\left(5\theta^4+47\theta^3+90\theta^2+47\theta+8\right)+2^{16} x^{3}\left(51\theta^4+192\theta^3+155\theta^2+48\theta+5\right)+2^{23} x^{4}\left((2\theta+1)^4\right)\)

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Coefficients of the holomorphic solution: 1, 16, 144, -70400, -9858800, ...
--> OEIS
Normalized instanton numbers (n0=1): 48, -1298, -9104, 387230, 102374160, ... ; Common denominator:...

Discriminant

\((32768z^2-208z+1)(1+64z)^2\)

Local exponents

\(0\)\(-\frac{ 1}{ 64}\)\(\infty\)\(\frac{ 13}{ 4096}+\frac{ 7}{ 4096}\sqrt{ 7}I\)\(\frac{ 13}{ 4096}-\frac{ 7}{ 4096}\sqrt{ 7}I\)\(s_1\)\(s_2\)
\(0\)\(0\)\(\frac{ 1}{ 2}\)\(0\)\(0\)\(0\)\(0\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)\(1\)
\(0\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)\(1\)
\(0\)\(4\)\(\frac{ 1}{ 2}\)\(2\)\(2\)\(2\)\(2\)

Note:

Sporadic Operator. There is a second MUM-point hiding at infinity,
corresponding to Operator AESZ 295/4.64

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16

New Number: 4.69 |  AESZ: 350  |  Superseeker: 49 173876/9  |  Hash: e6de16eb3758d2ed5687f4b2a2abf36b  

Degree: 4

\(\theta^4-x\left(24+184\theta+545\theta^2+722\theta^3+289\theta^4\right)+2^{3} 3 x^{2}\left(214\theta^4+2734\theta^3+4861\theta^2+2640\theta+468\right)+2^{6} 3^{2} x^{3}\left(1391\theta^4+5184\theta^3+4252\theta^2+1296\theta+126\right)+2^{10} 3^{6} x^{4}\left((2\theta+1)^4\right)\)

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Coefficients of the holomorphic solution: 1, 24, 1944, 232800, 34133400, ...
--> OEIS
Normalized instanton numbers (n0=1): 49, 136, 173876/9, 781152, 57087750, ... ; Common denominator:...

Discriminant

\((256z-1)(81z-1)(1+24z)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 81}\)\(\infty\)\(-\frac{ 1}{ 24}\)\(\frac{ 1}{ 256}\)
\(0\)\(0\)\(\frac{ 1}{ 2}\)\(0\)\(0\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(3\)\(1\)
\(0\)\(2\)\(\frac{ 1}{ 2}\)\(4\)\(2\)

Note:

Sporadic Operator. There is a second MUM-point hiding at infinity, corresponding to Operator AESZ 351/4.70

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17

New Number: 4.70 |  AESZ:  |  Superseeker: 177184 45194569320864  |  Hash: 56776b8a011d3f76a664ac5c7f492c1a  

Degree: 4

\(\theta^4+2^{4} x\left(22256\theta^4-38432\theta^3-23000\theta^2-3784\theta-321\right)+2^{18} 3^{3} x^{2}\left(1712\theta^4-18448\theta^3+8648\theta^2+2220\theta+279\right)-2^{30} 3^{6} x^{3}\left(4624\theta^4-2304\theta^3-1672\theta^2-576\theta-63\right)+2^{46} 3^{10} x^{4}\left((2\theta+1)^4\right)\)

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Coefficients of the holomorphic solution: 1, 5136, 98870544, 2900370796800, 105956691416931600, ...
--> OEIS
Normalized instanton numbers (n0=1): 177184, -1960034336, 45194569320864, -1351787074724461344, 47485667264376266736480, ... ; Common denominator:...

Discriminant

\((65536z-1)(20736z-1)(1+221184z)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 20736}\)\(\infty\)\(-\frac{ 1}{ 221184}\)\(\frac{ 1}{ 65536}\)
\(0\)\(0\)\(\frac{ 1}{ 2}\)\(0\)\(0\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(3\)\(1\)
\(0\)\(2\)\(\frac{ 1}{ 2}\)\(4\)\(2\)

Note:

Sporadic Operator. There is a second
MUM-point hiding at infinity, corresponding to
Operator AESZ 350/4.69

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18

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

Degree: 4

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

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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

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

Note:

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

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19

New Number: 4.73 |  AESZ: 362  |  Superseeker: -2656 -2493879008  |  Hash: 57a424b3b32b72260817cb8c45a8ae8f  

Degree: 4

\(\theta^4-2^{4} x\left(1088\theta^4-416\theta^3-212\theta^2-4\theta+3\right)+2^{12} 3^{3} x^{2}\left(928\theta^4-224\theta^3+448\theta^2+108\theta+9\right)-2^{20} 3^{6} x^{3}\left(320\theta^4+288\theta^3+220\theta^2+72\theta+9\right)+2^{28} 3^{10} x^{4}\left((2\theta+1)^4\right)\)

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Coefficients of the holomorphic solution: 1, 48, -40176, -103200000, -153639990000, ...
--> OEIS
Normalized instanton numbers (n0=1): -2656, -1985680, -2493879008, -3906525894360, -6910084057179168, ... ; Common denominator:...

Discriminant

\((5308416z^2-3584z+1)(-1+6912z)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 6912}\)\(\frac{ 7}{ 20736}+\frac{ 1}{ 5184}\sqrt{ 2}I\)\(\infty\)\(\frac{ 7}{ 20736}-\frac{ 1}{ 5184}\sqrt{ 2}I\)\(s_2\)\(s_1\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)\(0\)\(0\)\(0\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)
\(0\)\(3\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)
\(0\)\(4\)\(2\)\(\frac{ 1}{ 2}\)\(2\)\(2\)\(2\)

Note:

Sporadic Operator. There is a second MUM-point corresponding to Operator AESZ 361/4.72

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20

New Number: 7.4 |  AESZ:  |  Superseeker: -988/3 -14008436/3  |  Hash: 9bddfb88498c0a263be6ca541ae7e980  

Degree: 7

\(3^{2} \theta^4+2^{2} 3 x\left(760\theta^4+2048\theta^3+1423\theta^2+399\theta+42\right)-2^{7} x^{2}\left(20440\theta^4+25216\theta^3-4415\theta^2-4845\theta-795\right)+2^{12} x^{3}\left(39928\theta^4+16512\theta^3+23719\theta^2+11637\theta+1830\right)+2^{17} x^{4}\left(2928\theta^4-41856\theta^3-42871\theta^2-16873\theta-2425\right)+2^{23} x^{5}\left(608\theta^4+3968\theta^3+10676\theta^2+6177\theta+1089\right)+2^{29} x^{6}\left(272\theta^4+1056\theta^3+861\theta^2+264\theta+27\right)+2^{35} x^{7}\left((2\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -56, 21096, -12540800, 9146271400, ...
--> OEIS
Normalized instanton numbers (n0=1): -988/3, 57289/3, -14008436/3, 1385404666, -1599785191904/3, ... ; Common denominator:...

Discriminant

\((1+1248z-10240z^2+131072z^3)(-3+352z+2048z^2)^2\)

Local exponents

\(0\)\(-\frac{ 11}{ 128}-\frac{ 1}{ 128}\sqrt{ 145}\)\(-\frac{ 11}{ 128}+\frac{ 1}{ 128}\sqrt{ 145}\)\(\infty\) ≈\(0.039461+0.089595I\) ≈\(0.039461-0.089595I\) ≈\(-0.000796\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)\(0\)\(0\)\(0\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)
\(0\)\(3\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)
\(0\)\(4\)\(4\)\(\frac{ 1}{ 2}\)\(2\)\(2\)\(2\)

Note:

This is operator "7.4" from ...

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21

New Number: 7.5 |  AESZ:  |  Superseeker: 8096 9215266592  |  Hash: 90403d92f72b2839164c2bbd30933deb  

Degree: 7

\(\theta^4+2^{4} x\left(1088\theta^4-2048\theta^3-1260\theta^2-236\theta-19\right)+2^{15} x^{2}\left(1216\theta^4-5504\theta^3+11272\theta^2+3654\theta+423\right)+2^{24} x^{3}\left(11712\theta^4+190848\theta^3+97220\theta^2+27432\theta+2835\right)+2^{35} x^{4}\left(159712\theta^4+253376\theta^3+235372\theta^2+78648\theta+9491\right)-2^{46} x^{5}\left(81760\theta^4+62656\theta^3-46316\theta^2-33048\theta-5403\right)+2^{57} 3 x^{6}\left(3040\theta^4-2112\theta^3-2036\theta^2-528\theta-41\right)+2^{69} 3^{2} x^{7}\left((2\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 304, -113904, -2048902400, -4778502402800, ...
--> OEIS
Normalized instanton numbers (n0=1): 8096, -9179600, 9215266592, -8060820053720, 27014124083677664, ... ; Common denominator:...

Discriminant

\((2147483648z^3+40894464z^2-5120z+1)(-1-11264z+6291456z^2)^2\)

Local exponents

\(0\)\(\frac{ 11}{ 12288}-\frac{ 1}{ 12288}\sqrt{ 145}\)\(\infty\) ≈\(-0.019169\)\(\frac{ 11}{ 12288}+\frac{ 1}{ 12288}\sqrt{ 145}\) ≈\(6.3e-05-0.000143I\) ≈\(6.3e-05+0.000143I\)
\(0\)\(0\)\(\frac{ 1}{ 2}\)\(0\)\(0\)\(0\)\(0\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)\(1\)
\(0\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(3\)\(1\)\(1\)
\(0\)\(4\)\(\frac{ 1}{ 2}\)\(2\)\(4\)\(2\)\(2\)

Note:

This is operator "7.5" from ...

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22

New Number: 1.3 |  AESZ: 3  |  Superseeker: 32 26016  |  Hash: e7a9c334fb603aceccc0517dab63e7d4  

Degree: 1

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

Maple   LaTex

Coefficients of the holomorphic solution: 1, 16, 1296, 160000, 24010000, ...
--> OEIS
Normalized instanton numbers (n0=1): 32, 608, 26016, 1606496, 122373984, ... ; Common denominator:...

Discriminant

\(1-256z\)

Local exponents

\(0\)\(\infty\)\(\frac{ 1}{ 256}\)
\(0\)\(\frac{ 1}{ 2}\)\(0\)
\(0\)\(\frac{ 1}{ 2}\)\(1\)
\(0\)\(\frac{ 1}{ 2}\)\(1\)
\(0\)\(\frac{ 1}{ 2}\)\(2\)

Note:

A-incarnation: X(2,2,2,2) in P^7.

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