Summary

You searched for: Spectrum0=0,1/2,1/2,1

Your search produced 116 matches
 1-30  31-60  61-90  91-116 

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1

New Number: 2.17 |  AESZ: 111  |  Superseeker: 32 1440  |  Hash: d8535e0f3d0bfd4ebcc9c042df43c218  

Degree: 2

\(\theta^4-2^{4} x(2\theta+1)^2(8\theta^2+8\theta+3)+2^{12} x^{2}(2\theta+1)^2(2\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 48, 5904, 940800, 169520400, ...
--> OEIS
Normalized instanton numbers (n0=1): 32, -96, 1440, 19704, -14496, ... ; Common denominator:...

Discriminant

\((256z-1)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 256}\)\(\infty\)
\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(\frac{ 1}{ 2}\)\(\frac{ 1}{ 2}\)
\(0\)\(\frac{ 1}{ 2}\)\(\frac{ 3}{ 2}\)
\(0\)\(1\)\(\frac{ 3}{ 2}\)

Note:

B-Incarnations:
Fibre product 81111- x 18--21, 4*11-- x 53211,
Double Octics: D.O.8, D.O.36, D.O.73, D.O.249, D.O.258,D.O.265

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2

New Number: 2.18 |  AESZ: 110  |  Superseeker: 36 8076  |  Hash: 5060b638cac581d5f0f9dd7f40d90e6c  

Degree: 2

\(\theta^4-2^{2} 3 x(3\theta+1)(3\theta+2)(8\theta^2+8\theta+3)+2^{8} 3^{2} x^{2}(3\theta+1)(3\theta+2)(3\theta+4)(3\theta+5)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 72, 14760, 3951360, 1198751400, ...
--> OEIS
Normalized instanton numbers (n0=1): 36, -144, 8076, -57996, 6960672, ... ; Common denominator:...

Discriminant

\((432z-1)^2\)

Local exponents

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

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3

New Number: 2.19 |  AESZ: 112  |  Superseeker: -288 -96055968  |  Hash: 9a988f0cb0ca922885043cdadf98dd79  

Degree: 2

\(\theta^4-2^{4} 3 x(6\theta+1)(6\theta+5)(8\theta^2+8\theta+3)+2^{12} 3^{2} x^{2}(6\theta+1)(6\theta+5)(6\theta+7)(6\theta+11)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 720, 2273040, 9605756160, 46308725583120, ...
--> OEIS
Normalized instanton numbers (n0=1): -288, 162504, -96055968, 106571782296, -135291308081760, ... ; Common denominator:...

Discriminant

\((6912z-1)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 6912}\)\(\infty\)
\(0\)\(0\)\(\frac{ 1}{ 6}\)
\(0\)\(\frac{ 1}{ 2}\)\(\frac{ 5}{ 6}\)
\(0\)\(\frac{ 1}{ 2}\)\(\frac{ 7}{ 6}\)
\(0\)\(1\)\(\frac{ 11}{ 6}\)

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4

New Number: 3.10 |  AESZ: ~103  |  Superseeker: 10 664  |  Hash: 9239615e8ac132ca232c13367a39ae3b  

Degree: 3

\(\theta^4-2 x\left(86\theta^4+172\theta^3+143\theta^2+57\theta+9\right)+2^{2} 3^{2} x^{2}(\theta+1)^2(236\theta^2+472\theta+187)-2^{4} 3^{4} 5^{2} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 18, 630, 28980, 1593270, ...
--> OEIS
Normalized instanton numbers (n0=1): 10, 24, 664, 9088, 234388, ... ; Common denominator:...

Discriminant

\(-(100z-1)(-1+36z)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 100}\)\(\frac{ 1}{ 36}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(2\)
\(0\)\(2\)\(1\)\(\frac{ 5}{ 2}\)

Note:

Operator equivalent to AESZ 103 =$c \ast c$.

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5

New Number: 3.11 |  AESZ:  |  Superseeker: 37 15270  |  Hash: e7db0935aa1b331d8fb696a009d2d7bb  

Degree: 3

\(\theta^4-x\left(865\theta^4+1730\theta^3+1501\theta^2+636\theta+108\right)+2^{5} 3^{2} x^{2}(\theta+1)^2(866\theta^2+1732\theta+709)-2^{8} 3^{4} 17^{2} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 108, 19908, 4278240, 990152100, ...
--> OEIS
Normalized instanton numbers (n0=1): 37, -570, 15270, -529994, 21300463, ... ; Common denominator:...

Discriminant

\(-(289z-1)(-1+288z)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 289}\)\(\frac{ 1}{ 288}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(2\)
\(0\)\(2\)\(1\)\(\frac{ 5}{ 2}\)

Note:

Operator equivalent to AESZ 144=c \ast c$

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6

New Number: 3.2 |  AESZ: 227  |  Superseeker: -900 8364884  |  Hash: 2e00a51fe0c232d13a452380f44c79da  

Degree: 3

\(\theta^4+2^{2} 3^{2} x\left(132\theta^4+264\theta^3+201\theta^2+69\theta+10\right)+2^{9} 3^{6} x^{2}\left(20\theta^4+80\theta^3+107\theta^2+54\theta+10\right)+2^{12} 3^{10} x^{3}(2\theta+5)^2(2\theta+1)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -360, 314280, -348076800, 431342188200, ...
--> OEIS
Normalized instanton numbers (n0=1): -900, -27387, 8364884, 2066389488, -208833104160, ... ; Common denominator:...

Discriminant

\((1296z+1)(1+1728z)^2\)

Local exponents

\(-\frac{ 1}{ 1296}\)\(-\frac{ 1}{ 1728}\)\(0\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(\frac{ 1}{ 2}\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(\frac{ 1}{ 2}\)\(0\)\(\frac{ 5}{ 2}\)
\(2\)\(1\)\(0\)\(\frac{ 5}{ 2}\)

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7

New Number: 3.34 |  AESZ:  |  Superseeker: 16 1744  |  Hash: 931a876bfe4d4aa192c6e18e74047640  

Degree: 3

\(\theta^4-2^{4} x\left(25\theta^4+50\theta^3+43\theta^2+18\theta+3\right)+2^{11} x^{2}(26\theta^2+52\theta+21)(\theta+1)^2-2^{16} 3^{2} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 48, 3984, 387840, 40818960, ...
--> OEIS
Normalized instanton numbers (n0=1): 16, -110, 1744, -29526, 644016, ... ; Common denominator:...

Discriminant

\(-(144z-1)(-1+128z)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 144}\)\(\frac{ 1}{ 128}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(2\)
\(0\)\(2\)\(1\)\(\frac{ 5}{ 2}\)

Note:

Operator equivalent to AESZ 107 $=d \ast d$

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8

New Number: 3.3 |  AESZ: 228  |  Superseeker: -68 -18628/3  |  Hash: b15f49e2c20021dbc50eaf05a6fd3126  

Degree: 3

\(\theta^4+2^{2} x\left(176\theta^4+352\theta^3+289\theta^2+113\theta+18\right)+2^{11} x^{2}\left(80\theta^4+320\theta^3+449\theta^2+258\theta+54\right)+2^{16} 3 x^{3}(2\theta+5)(2\theta+1)(4\theta+3)(4\theta+9)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -72, 10152, -1739520, 327839400, ...
--> OEIS
Normalized instanton numbers (n0=1): -68, -835, -18628/3, 359052, 23710944, ... ; Common denominator:...

Discriminant

\((192z+1)(1+256z)^2\)

Local exponents

\(-\frac{ 1}{ 192}\)\(-\frac{ 1}{ 256}\)\(0\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(\frac{ 1}{ 2}\)\(0\)\(\frac{ 3}{ 4}\)
\(1\)\(\frac{ 1}{ 2}\)\(0\)\(\frac{ 9}{ 4}\)
\(2\)\(1\)\(0\)\(\frac{ 5}{ 2}\)

Note:

This is operator "3.3" from ...

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9

New Number: 3.4 |  AESZ:  |  Superseeker: -9 -748  |  Hash: 350ef7c6e038467a3f50bfbe164fa73a  

Degree: 3

\(\theta^4+3^{2} x\left(33\theta^4+66\theta^3+57\theta^2+24\theta+4\right)+2^{3} 3^{6} x^{2}(\theta+1)^2(5\theta^2+10\theta+4)+2^{2} 3^{10} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -36, 2268, -168840, 13664700, ...
--> OEIS
Normalized instanton numbers (n0=1): -9, -279/4, -748, -9612, -155448, ... ; Common denominator:...

Discriminant

\((81z+1)(1+108z)^2\)

Local exponents

\(-\frac{ 1}{ 81}\)\(-\frac{ 1}{ 108}\)\(0\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(\frac{ 1}{ 2}\)\(0\)\(1\)
\(1\)\(\frac{ 1}{ 2}\)\(0\)\(2\)
\(2\)\(1\)\(0\)\(\frac{ 5}{ 2}\)

Note:

Operator equivalent to AESZ 165= $f \ast f$.

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10

New Number: 3.8 |  AESZ: ~100  |  Superseeker: 5 454  |  Hash: 82a1ac6ac6fb9ab2e4d6b5d5790d1d9b  

Degree: 3

\(\theta^4+x\left(15\theta^4+30\theta^3+35\theta^2+20\theta+4\right)-2^{5} x^{2}(\theta+1)^2(66\theta^2+132\theta+53)-2^{8} 7^{2} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -4, 132, -1120, 72100, ...
--> OEIS
Normalized instanton numbers (n0=1): 5, 42, 454, 7498, 154351, ... ; Common denominator:...

Discriminant

\(-(49z-1)(1+32z)^2\)

Local exponents

\(-\frac{ 1}{ 32}\)\(0\)\(\frac{ 1}{ 49}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(\frac{ 1}{ 2}\)\(0\)\(1\)\(1\)
\(\frac{ 1}{ 2}\)\(0\)\(1\)\(2\)
\(1\)\(0\)\(2\)\(\frac{ 5}{ 2}\)

Note:

Operator equivalent to AESZ 100= $ a \ast a$

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11

New Number: 3.9 |  AESZ: ~101  |  Superseeker: 13 2650  |  Hash: a6878d847acf199583e8168a33967174  

Degree: 3

\(\theta^4-x\left(113\theta^4+226\theta^3+173\theta^2+60\theta+8\right)-2^{3} x^{2}(\theta+1)^2(119\theta^2+238\theta+92)-2^{2} 11^{2} x^{3}(\theta+1)(\theta+2)(2\theta+1)(2\theta+5)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 8, 336, 19880, 1420720, ...
--> OEIS
Normalized instanton numbers (n0=1): 13, 128, 2650, 79400, 2921395, ... ; Common denominator:...

Discriminant

\(-(121z-1)(4z+1)^2\)

Local exponents

\(-\frac{ 1}{ 4}\)\(0\)\(\frac{ 1}{ 121}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(\frac{ 1}{ 2}\)\(0\)\(1\)\(1\)
\(\frac{ 1}{ 2}\)\(0\)\(1\)\(2\)
\(1\)\(0\)\(2\)\(\frac{ 5}{ 2}\)

Note:

Operator equivalent to $AESZ 101=$b \ast b$.

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12

New Number: 4.55 |  AESZ: 276  |  Superseeker: -188832 -101990911789344  |  Hash: 797e27181bf0a060708a3d221ec79699  

Degree: 4

\(\theta^4-2^{4} 3 x\left(18432\theta^4-4608\theta^3-1024\theta^2+1280\theta+221\right)+2^{17} 3^{4} x^{2}\left(25344\theta^4-2304\theta^3+11680\theta^2+1472\theta-33\right)-2^{28} 3^{8} x^{3}\left(18432\theta^4+13824\theta^3+11392\theta^2+3264\theta+359\right)+2^{46} 3^{12} x^{4}(3\theta+1)(2\theta+1)^2(3\theta+2)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 10608, 477012240, 30101658720000, 2213759644568010000, ...
--> OEIS
Normalized instanton numbers (n0=1): -188832, -3134817768, -101990911789344, -4414817659429205136, -223930278487379610386400, ... ; Common denominator:...

Discriminant

\((331776z-1)^2(110592z-1)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 331776}\)\(\frac{ 1}{ 110592}\)\(\infty\)
\(0\)\(0\)\(0\)\(\frac{ 1}{ 3}\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(\frac{ 1}{ 2}\)
\(0\)\(3\)\(\frac{ 1}{ 2}\)\(\frac{ 1}{ 2}\)
\(0\)\(4\)\(1\)\(\frac{ 2}{ 3}\)

Note:

Sporadic Operator.

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13

New Number: 4.77 |  AESZ:  |  Superseeker: 1 68/9  |  Hash: f9623221ffe8be4c1e31a6e6ce195a37  

Degree: 4

\(\theta^4-x\left(16+80\theta+161\theta^2+162\theta^3+81\theta^4\right)+2^{3} x^{2}\left(303\theta^4+1212\theta^3+1952\theta^2+1480\theta+440\right)-2^{6} x^{3}(124\theta^2+372\theta+263)(2\theta+3)^2+2^{9} 3 5^{2} x^{4}(\theta+1)(2\theta+5)(2\theta+3)(\theta+3)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 16, 280, 5152, 98200, ...
--> OEIS
Normalized instanton numbers (n0=1): 1, -2, 68/9, -30, 150, ... ; Common denominator:...

Discriminant

\((25z-1)(24z-1)(-1+16z)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 25}\)\(\frac{ 1}{ 24}\)\(\frac{ 1}{ 16}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(1\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(\frac{ 3}{ 2}\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(\frac{ 5}{ 2}\)
\(0\)\(2\)\(2\)\(1\)\(3\)

Note:

Sporadic Operator.
B-Incarnation: SII4411

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14

New Number: 5.109 |  AESZ: 373  |  Superseeker: 50 68472  |  Hash: 8b0fbfc0016c3fb02fd42d4ff919e0f8  

Degree: 5

\(\theta^4-2 x\left(190\theta^4+308\theta^3+227\theta^2+73\theta+9\right)+2^{2} x^{2}\left(4780\theta^4+6304\theta^3+2395\theta^2+642\theta+135\right)-2^{4} 3 x^{3}\left(6700\theta^4+8472\theta^3+7607\theta^2+3615\theta+648\right)+2^{7} 3^{2} x^{4}(2\theta+1)(760\theta^3+1464\theta^2+1211\theta+375)-2^{10} 3^{6} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 18, 1782, 276660, 52396470, ...
--> OEIS
Normalized instanton numbers (n0=1): 50, 1299, 68472, 5536032, 555252324, ... ; Common denominator:...

Discriminant

\(-(-1+324z)(24z-1)^2(4z-1)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 324}\)\(\frac{ 1}{ 24}\)\(\frac{ 1}{ 4}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)
\(0\)\(1\)\(3\)\(\frac{ 1}{ 2}\)\(1\)
\(0\)\(2\)\(4\)\(1\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.109" from ...

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15

New Number: 5.111 |  AESZ: 380  |  Superseeker: 12 2320  |  Hash: 85214e3836a67470a05358a4d38fb124  

Degree: 5

\(\theta^4-2 x\left(60\theta^4+90\theta^3+68\theta^2+23\theta+3\right)+2^{2} x^{2}\left(313\theta^4-398\theta^3-1417\theta^2-1033\theta-252\right)+2^{3} x^{3}\left(654\theta^4+5064\theta^3+3574\theta^2+129\theta-405\right)-2^{4} 5 x^{4}\left(628\theta^4-40\theta^3-1699\theta^2-1661\theta-480\right)-2^{6} 3 5^{2} x^{5}(\theta+1)^2(6\theta+5)(6\theta+7)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 6, 246, 13020, 832950, ...
--> OEIS
Normalized instanton numbers (n0=1): 12, 511/4, 2320, 63507, 2180312, ... ; Common denominator:...

Discriminant

\(-(108z-1)(4z+1)^2(10z-1)^2\)

Local exponents

\(-\frac{ 1}{ 4}\)\(0\)\(\frac{ 1}{ 108}\)\(\frac{ 1}{ 10}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 5}{ 6}\)
\(\frac{ 1}{ 2}\)\(0\)\(1\)\(1\)\(1\)
\(\frac{ 1}{ 2}\)\(0\)\(1\)\(3\)\(1\)
\(1\)\(0\)\(2\)\(4\)\(\frac{ 7}{ 6}\)

Note:

This is operator "5.111" from ...

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16

New Number: 5.114 |  AESZ: 412  |  Superseeker: -1312 -127846048  |  Hash: 4a9870395db313fa368bbafcfc6a7435  

Degree: 5

\(\theta^4-2^{4} x\left(560\theta^4-32\theta^3+56\theta^2+72\theta+15\right)+2^{15} x^{2}\left(896\theta^4+272\theta^3+604\theta^2+196\theta+21\right)-2^{24} 3^{2} x^{3}\left(288\theta^4+352\theta^3+364\theta^2+164\theta+29\right)+2^{35} 3^{3} x^{4}(2\theta+1)(16\theta^3+32\theta^2+28\theta+9)-2^{46} 3^{3} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 240, 118032, 72810240, 50454043920, ...
--> OEIS
Normalized instanton numbers (n0=1): -1312, -301048, -127846048, -70845744192, -45645879602784, ... ; Common denominator:...

Discriminant

\(-(-1+768z)(3072z-1)^2(1024z-1)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 3072}\)\(\frac{ 1}{ 1024}\)\(\frac{ 1}{ 768}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(1\)
\(0\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(1\)
\(0\)\(4\)\(1\)\(2\)\(\frac{ 3}{ 2}\)

Note:

B-Incarnation as double octic D.O.256

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17

New Number: 5.115 |  AESZ: 413  |  Superseeker: -3843 -2715123387  |  Hash: 2cc16ba9e49744872ae72bfd6b36d064  

Degree: 5

\(\theta^4-3^{2} x\left(2835\theta^4-162\theta^3+261\theta^2+342\theta+68\right)+2^{2} 3^{9} x^{2}\left(3024\theta^4+918\theta^3+1977\theta^2+606\theta+64\right)-2^{2} 3^{16} x^{3}\left(5832\theta^4+7128\theta^3+7137\theta^2+3087\theta+524\right)+2^{4} 3^{25} x^{4}(2\theta+1)(72\theta^3+144\theta^2+121\theta+36)-2^{6} 3^{31} x^{5}(2\theta+1)(3\theta+2)(3\theta+4)(2\theta+3)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 612, 836244, 1455469200, 2860801391700, ...
--> OEIS
Normalized instanton numbers (n0=1): -3843, -9668061/4, -2715123387, -3984527414448, -6798579266503881, ... ; Common denominator:...

Discriminant

\(-(-1+2187z)(8748z-1)^2(2916z-1)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 8748}\)\(\frac{ 1}{ 2916}\)\(\frac{ 1}{ 2187}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(\frac{ 2}{ 3}\)
\(0\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(\frac{ 4}{ 3}\)
\(0\)\(4\)\(1\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.115" from ...

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18

New Number: 5.116 |  AESZ: 414  |  Superseeker: -22432 -425234532128  |  Hash: 973fceefe183415b5d0e15e5a0bd12f5  

Degree: 5

\(\theta^4-2^{4} x\left(8960\theta^4-512\theta^3+736\theta^2+992\theta+183\right)+2^{19} x^{2}\left(14336\theta^4+4352\theta^3+9008\theta^2+2544\theta+261\right)-2^{32} 3^{2} x^{3}\left(4608\theta^4+5632\theta^3+5408\theta^2+2208\theta+351\right)+2^{49} 3^{3} x^{4}(2\theta+1)^2(32\theta^2+48\theta+27)-2^{64} 3^{3} x^{5}(2\theta+1)^2(2\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 2928, 21778704, 210543916800, 2314156512099600, ...
--> OEIS
Normalized instanton numbers (n0=1): -22432, -74752296, -425234532128, -3159114140624208, -27288043319514722784, ... ; Common denominator:...

Discriminant

\(-(-1+12288z)(49152z-1)^2(16384z-1)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 49152}\)\(\frac{ 1}{ 16384}\)\(\frac{ 1}{ 12288}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(\frac{ 1}{ 2}\)
\(0\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(\frac{ 3}{ 2}\)
\(0\)\(4\)\(1\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.116" from ...

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19

New Number: 5.117 |  AESZ: 415  |  Superseeker: -1083168 -32204207145918624  |  Hash: 480ebcdf255c1fe6ad1cac7896e482fa  

Degree: 5

\(\theta^4-2^{4} 3^{2} x\left(45360\theta^4-2592\theta^3+3096\theta^2+4392\theta+695\right)+2^{15} 3^{9} x^{2}\left(24192\theta^4+7344\theta^3+14340\theta^2+3516\theta+335\right)-2^{24} 3^{16} x^{3}\left(23328\theta^4+28512\theta^3+25740\theta^2+9540\theta+1325\right)+2^{35} 3^{25} x^{4}(2\theta+1)(144\theta^3+288\theta^2+212\theta+45)-2^{46} 3^{31} x^{5}(2\theta+1)(3\theta+1)(3\theta+5)(2\theta+3)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 100080, 32388472080, 14040210456518400, 6986717866758049635600, ...
--> OEIS
Normalized instanton numbers (n0=1): -1083168, -148187321784, -32204207145918624, -9094164085684648886400, -2986312705706358630596895840, ... ; Common denominator:...

Discriminant

\(-(-1+559872z)(2239488z-1)^2(746496z-1)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 2239488}\)\(\frac{ 1}{ 746496}\)\(\frac{ 1}{ 559872}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 3}\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(\frac{ 1}{ 2}\)
\(0\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(\frac{ 3}{ 2}\)
\(0\)\(4\)\(1\)\(2\)\(\frac{ 5}{ 3}\)

Note:

This is operator "5.117" from ...

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20

New Number: 5.118 |  AESZ: 416  |  Superseeker: 1312 127846048  |  Hash: 3ae3241981d64d9c9cc38b29974fa202  

Degree: 5

\(\theta^4+2^{4} x\left(560\theta^4-32\theta^3+56\theta^2+72\theta+15\right)+2^{15} x^{2}\left(896\theta^4+272\theta^3+604\theta^2+196\theta+21\right)+2^{24} 3^{2} x^{3}\left(288\theta^4+352\theta^3+364\theta^2+164\theta+29\right)+2^{35} 3^{3} x^{4}(2\theta+1)(16\theta^3+32\theta^2+28\theta+9)+2^{46} 3^{3} x^{5}(2\theta+1)(\theta+1)^2(2\theta+3)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -240, 118032, -72810240, 50454043920, ...
--> OEIS
Normalized instanton numbers (n0=1): 1312, -301376, 127846048, -70845744192, 45645879602784, ... ; Common denominator:...

Discriminant

\((1+768z)(1024z+1)^2(3072z+1)^2\)

Local exponents

\(-\frac{ 1}{ 768}\)\(-\frac{ 1}{ 1024}\)\(-\frac{ 1}{ 3072}\)\(0\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(\frac{ 1}{ 2}\)\(1\)\(0\)\(1\)
\(1\)\(\frac{ 1}{ 2}\)\(3\)\(0\)\(1\)
\(2\)\(1\)\(4\)\(0\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.118" from ...

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21

New Number: 5.12 |  AESZ: 74  |  Superseeker: -30 -14632  |  Hash: e668180adb7c88d4e5fbab5eb7ee61c7  

Degree: 5

\(\theta^4-2 3 x\left(99\theta^4+36\theta^3+39\theta^2+21\theta+4\right)+2^{2} 3^{2} x^{2}\left(3807\theta^4+3564\theta^3+3798\theta^2+1683\theta+284\right)-2^{3} 3^{5} x^{3}\left(7857\theta^4+13608\theta^3+14562\theta^2+7317\theta+1444\right)+2^{4} 3^{9} x^{4}\left(2592\theta^4+7128\theta^3+8550\theta^2+4851\theta+1052\right)-2^{5} 3^{13} x^{5}(3\theta+2)(3\theta+4)(6\theta+5)(6\theta+7)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 24, 1152, 71520, 5101200, ...
--> OEIS
Normalized instanton numbers (n0=1): -30, -516, -14632, -4227807/8, -22139868, ... ; Common denominator:...

Discriminant

\(-(-1+54z)(162z-1)^2(108z-1)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 162}\)\(\frac{ 1}{ 108}\)\(\frac{ 1}{ 54}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 2}{ 3}\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(\frac{ 5}{ 6}\)
\(0\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(\frac{ 7}{ 6}\)
\(0\)\(4\)\(1\)\(2\)\(\frac{ 4}{ 3}\)

Note:

This is operator "5.12" from ...

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22

New Number: 5.13 |  AESZ: 83  |  Superseeker: -80 -174096  |  Hash: 171e1251d8e4f7de878d0d07de6f58ab  

Degree: 5

\(\theta^4-2^{4} x\left(88\theta^4+32\theta^3+33\theta^2+17\theta+3\right)+2^{9} x^{2}\left(1504\theta^4+1408\theta^3+1436\theta^2+596\theta+93\right)-2^{18} x^{3}\left(776\theta^4+1344\theta^3+1381\theta^2+651\theta+117\right)+2^{23} 3 x^{4}(2\theta+1)(512\theta^3+1152\theta^2+1054\theta+339)-2^{31} 3^{2} x^{5}(2\theta+1)(4\theta+3)(4\theta+5)(2\theta+3)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 48, 5328, 779520, 131619600, ...
--> OEIS
Normalized instanton numbers (n0=1): -80, -2954, -174096, -13270953, -1179175536, ... ; Common denominator:...

Discriminant

\(-(128z-1)(384z-1)^2(256z-1)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 384}\)\(\frac{ 1}{ 256}\)\(\frac{ 1}{ 128}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(\frac{ 3}{ 4}\)
\(0\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(\frac{ 5}{ 4}\)
\(0\)\(4\)\(1\)\(2\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.13" from ...

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23

New Number: 5.14 |  AESZ: 116  |  Superseeker: 64 23360  |  Hash: 0b366ad8c78b6697205c5a7fff270f5b  

Degree: 5

\(\theta^4-2^{5} x\left(10\theta^4+26\theta^3+20\theta^2+7\theta+1\right)+2^{8} x^{2}\left(52\theta^4+472\theta^3+832\theta^2+492\theta+103\right)+2^{16} x^{3}\left(14\theta^4+12\theta^3-96\theta^2-105\theta-29\right)-2^{18} x^{4}(2\theta+1)(56\theta^3+468\theta^2+646\theta+249)-2^{24} x^{5}(2\theta+1)(4\theta+3)(4\theta+5)(2\theta+3)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 32, 2448, 273920, 38525200, ...
--> OEIS
Normalized instanton numbers (n0=1): 64, 12, 23360, 654490, 53956288, ... ; Common denominator:...

Discriminant

\(-(-1+256z)(32z+1)^2(64z-1)^2\)

Local exponents

\(-\frac{ 1}{ 32}\)\(0\)\(\frac{ 1}{ 256}\)\(\frac{ 1}{ 64}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(0\)\(1\)\(\frac{ 1}{ 2}\)\(\frac{ 3}{ 4}\)
\(3\)\(0\)\(1\)\(\frac{ 1}{ 2}\)\(\frac{ 5}{ 4}\)
\(4\)\(0\)\(2\)\(1\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.14" from ...

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24

New Number: 5.19 |  AESZ: 180  |  Superseeker: -624 -43406256  |  Hash: c174fb2dfd87730e48b4ae8b57ac66df  

Degree: 5

\(\theta^4-2^{4} 3 x\left(198\theta^4+72\theta^3+69\theta^2+33\theta+5\right)+2^{9} 3^{2} x^{2}\left(7614\theta^4+7128\theta^3+6813\theta^2+2529\theta+340\right)-2^{14} 3^{5} x^{3}\left(15714\theta^4+27216\theta^3+26343\theta^2+11151\theta+1685\right)+2^{19} 3^{9} x^{4}(3\theta+1)(3\theta+2)(576\theta^2+1008\theta+605)-2^{27} 3^{13} x^{5}(3\theta+1)(3\theta+2)(3\theta+4)(3\theta+5)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 240, 173520, 170016000, 193451504400, ...
--> OEIS
Normalized instanton numbers (n0=1): -624, -137190, -43406256, -18281817141, -9083828410320, ... ; Common denominator:...

Discriminant

\(-(-1+864z)(2592z-1)^2(1728z-1)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 2592}\)\(\frac{ 1}{ 1728}\)\(\frac{ 1}{ 864}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 3}\)
\(0\)\(1\)\(\frac{ 1}{ 2}\)\(1\)\(\frac{ 2}{ 3}\)
\(0\)\(3\)\(\frac{ 1}{ 2}\)\(1\)\(\frac{ 4}{ 3}\)
\(0\)\(4\)\(1\)\(2\)\(\frac{ 5}{ 3}\)

Note:

This is operator "5.19" from ...

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25

New Number: 5.23 |  AESZ: 194  |  Superseeker: 126/17 11700/17  |  Hash: 6bf19665aa6705f30ef88df42bc4eac4  

Degree: 5

\(17^{2} \theta^4-17 x\left(1465\theta^4+2768\theta^3+2200\theta^2+816\theta+119\right)+2 x^{2}\left(62015\theta^4+131582\theta^3+125017\theta^2+65926\theta+15300\right)-2 3^{3} x^{3}\left(4325\theta^4+10914\theta^3+12803\theta^2+7446\theta+1700\right)+3^{6} x^{4}\left(265\theta^4+836\theta^3+1118\theta^2+700\theta+168\right)-3^{10} x^{5}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 7, 183, 7225, 345079, ...
--> OEIS
Normalized instanton numbers (n0=1): 126/17, 848/17, 11700/17, 229808/17, 5539258/17, ... ; Common denominator:...

Discriminant

\(-(-1+81z)(27z-17)^2(z-1)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 81}\)\(\frac{ 17}{ 27}\)\(1\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(1\)
\(0\)\(1\)\(1\)\(\frac{ 1}{ 2}\)\(1\)
\(0\)\(1\)\(3\)\(\frac{ 1}{ 2}\)\(1\)
\(0\)\(2\)\(4\)\(1\)\(1\)

Note:

There is a second MUM-point at infinity, corresponding to Operator AESZ 199/5.26

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26

New Number: 5.26 |  AESZ: 199  |  Superseeker: -2 3820/9  |  Hash: f7b5c9e3ad50b0885d03c98d07a051f1  

Degree: 5

\(\theta^4-x\left(15+88\theta+200\theta^2+224\theta^3+265\theta^4\right)+2 3 x^{2}\left(4325\theta^4+6386\theta^3+6011\theta^2+2718\theta+468\right)-2 3^{2} x^{3}\left(62015\theta^4+116478\theta^3+102361\theta^2+37422\theta+4824\right)+3^{6} 17 x^{4}\left(1465\theta^4+3092\theta^3+2686\theta^2+1140\theta+200\right)-3^{10} 17^{2} x^{5}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 15, 567, 28113, 1584279, ...
--> OEIS
Normalized instanton numbers (n0=1): -2, 28, 3820/9, 3924, 21606, ... ; Common denominator:...

Discriminant

\(-(z-1)(81z-1)^2(51z-1)^2\)

Local exponents

\(0\)\(\frac{ 1}{ 81}\)\(\frac{ 1}{ 51}\)\(1\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(1\)
\(0\)\(\frac{ 1}{ 2}\)\(1\)\(1\)\(1\)
\(0\)\(\frac{ 1}{ 2}\)\(3\)\(1\)\(1\)
\(0\)\(1\)\(4\)\(2\)\(1\)

Note:

There is a second MUM-point at infinity, corresponding to
Operator AESZ 194/5.23.

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27

New Number: 5.45 |  AESZ: 242  |  Superseeker: -18 1568/3  |  Hash: 562c18d54c0080ebb0bb01b14a8241ce  

Degree: 5

\(\theta^4+2 3 x\left(72\theta^4+108\theta^3+91\theta^2+37\theta+6\right)+2^{2} 3^{3} x^{2}\left(648\theta^4+1800\theta^3+2211\theta^2+1248\theta+260\right)+2^{4} 3^{5} x^{3}\left(1344\theta^4+4968\theta^3+7320\theta^2+4749\theta+1072\right)+2^{6} 3^{7} x^{4}(2\theta+1)(630\theta^3+2241\theta^2+2617\theta+971)+2^{8} 3^{10} x^{5}(2\theta+1)(6\theta+5)(6\theta+7)(2\theta+3)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -36, 2484, -208080, 19221300, ...
--> OEIS
Normalized instanton numbers (n0=1): -18, 99/2, 1568/3, 22698, -165960, ... ; Common denominator:...

Discriminant

\((1+144z)(36z+1)^2(108z+1)^2\)

Local exponents

\(-\frac{ 1}{ 36}\)\(-\frac{ 1}{ 108}\)\(-\frac{ 1}{ 144}\)\(0\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 1}{ 2}\)
\(1\)\(\frac{ 1}{ 2}\)\(1\)\(0\)\(\frac{ 5}{ 6}\)
\(3\)\(\frac{ 1}{ 2}\)\(1\)\(0\)\(\frac{ 7}{ 6}\)
\(4\)\(1\)\(2\)\(0\)\(\frac{ 3}{ 2}\)

Note:

This is operator "5.45" from ...

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28

New Number: 5.47 |  AESZ: 246  |  Superseeker: -4/5 -108/5  |  Hash: f51a0c39f9179dc6a561b9afb6f9d85f  

Degree: 5

\(5^{2} \theta^4-2^{2} 5 x\left(12\theta^4+48\theta^3+49\theta^2+25\theta+5\right)-2^{4} x^{2}\left(544\theta^4+1792\theta^3+2444\theta^2+1580\theta+405\right)+2^{9} x^{3}\left(112\theta^4+960\theta^3+2306\theta^2+2130\theta+685\right)+2^{12} x^{4}\left(144\theta^4+768\theta^3+1308\theta^2+924\theta+235\right)+2^{20} x^{5}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 4, 44, 400, 5356, ...
--> OEIS
Normalized instanton numbers (n0=1): -4/5, 22/5, -108/5, 694/5, -1040, ... ; Common denominator:...

Discriminant

\((1+16z)(16z+5)^2(16z-1)^2\)

Local exponents

\(-\frac{ 5}{ 16}\)\(-\frac{ 1}{ 16}\)\(0\)\(\frac{ 1}{ 16}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(0\)\(\frac{ 1}{ 2}\)\(1\)
\(3\)\(1\)\(0\)\(\frac{ 1}{ 2}\)\(1\)
\(4\)\(2\)\(0\)\(1\)\(1\)

Note:

There is a second MUM-point at infinity,
corresponding to Operator AESZ 247/5.48

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29

New Number: 5.48 |  AESZ: 247  |  Superseeker: 608 22293216  |  Hash: 6c0503129f3500c26cf001c1908a17f7  

Degree: 5

\(\theta^4+2^{4} x\left(144\theta^4-192\theta^3-132\theta^2-36\theta-5\right)+2^{13} x^{2}\left(112\theta^4-512\theta^3+98\theta^2+50\theta+13\right)-2^{20} x^{3}\left(544\theta^4+384\theta^3+332\theta^2+108\theta+21\right)-2^{30} 5 x^{4}\left(12\theta^4-23\theta^2-23\theta-7\right)+2^{40} 5^{2} x^{5}\left((\theta+1)^4\right)\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, 80, 11024, 1850624, 343952656, ...
--> OEIS
Normalized instanton numbers (n0=1): 608, -85544, 22293216, -7629059800, 3042437418016, ... ; Common denominator:...

Discriminant

\((1+256z)(1280z+1)^2(256z-1)^2\)

Local exponents

\(-\frac{ 1}{ 256}\)\(-\frac{ 1}{ 1280}\)\(0\)\(\frac{ 1}{ 256}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(1\)
\(1\)\(1\)\(0\)\(\frac{ 1}{ 2}\)\(1\)
\(1\)\(3\)\(0\)\(\frac{ 1}{ 2}\)\(1\)
\(2\)\(4\)\(0\)\(1\)\(1\)

Note:

There is a second MUM-point at infinity,
corresponding to Operaor AESZ 246/ 5.47

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30

New Number: 5.50 |  AESZ: 249  |  Superseeker: -44/5 -596  |  Hash: 85592af20bbb190e37428e945664c2f3  

Degree: 5

\(5^{2} \theta^4+2^{2} 5 x\left(148\theta^4+392\theta^3+341\theta^2+145\theta+25\right)+2^{4} x^{2}\left(4096\theta^4+32128\theta^3+57016\theta^2+37920\theta+9175\right)-2^{8} x^{3}\left(6656\theta^4+7680\theta^3-36960\theta^2-49920\theta-16985\right)-2^{15} x^{4}\left(512\theta^4+4864\theta^3+9136\theta^2+6464\theta+1587\right)+2^{20} x^{5}(4\theta+5)^2(4\theta+3)^2\)

Maple   LaTex

Coefficients of the holomorphic solution: 1, -20, 684, -28496, 1317100, ...
--> OEIS
Normalized instanton numbers (n0=1): -44/5, -277/5, -596, -7236, -502128/5, ... ; Common denominator:...

Discriminant

\((1+16z)(64z+1)^2(64z-5)^2\)

Local exponents

\(-\frac{ 1}{ 16}\)\(-\frac{ 1}{ 64}\)\(0\)\(\frac{ 5}{ 64}\)\(\infty\)
\(0\)\(0\)\(0\)\(0\)\(\frac{ 3}{ 4}\)
\(1\)\(\frac{ 1}{ 2}\)\(0\)\(1\)\(\frac{ 3}{ 4}\)
\(1\)\(\frac{ 1}{ 2}\)\(0\)\(3\)\(\frac{ 5}{ 4}\)
\(2\)\(1\)\(0\)\(4\)\(\frac{ 5}{ 4}\)

Note:

This is operator "5.50" from ...

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