### Summary

You searched for: inst=12

1

New Number: 2.1 |  AESZ: 45  |  Superseeker: 12 3204  |  Hash: cdf289f6febf84eb577a238542a57457

Degree: 2

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

Maple   LaTex

Coefficients of the holomorphic solution: 1, 8, 360, 22400, 1695400, ...
--> OEIS
Normalized instanton numbers (n0=1): 12, 163, 3204, 107582, 4203360, ... ; Common denominator:...

#### Discriminant

$-(16z+1)(128z-1)$

#### Local exponents

$-\frac{ 1}{ 16}$$0$$\frac{ 1}{ 128}$$\infty$
$0$$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 $A \ast a$, where $A$ is (:case 2.1.1)

2

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

3

New Number: 3.25 |  AESZ:  |  Superseeker: -2 -308/3  |  Hash: 287da3a26b0da679d81da411b46958d1

Degree: 3

$\theta^4+2 x(2\theta+1)^2(7\theta^2+7\theta+3)+2^{2} x^{2}(2\theta+1)(2\theta+3)(29\theta^2+58\theta+33)+2^{4} 3 5 x^{3}(2\theta+1)(2\theta+3)^2(2\theta+5)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, -6, 90, -2100, 59850, ...
--> OEIS
Normalized instanton numbers (n0=1): -2, 12, -308/3, 1058, -71158/5, ... ; Common denominator:...

#### Discriminant

$(48z+1)(80z^2+8z+1)$

#### Local exponents

$-\frac{ 1}{ 20}-\frac{ 1}{ 10}I$$-\frac{ 1}{ 20}+\frac{ 1}{ 10}I$$-\frac{ 1}{ 48}$$0$$\infty$
$0$$0$$0$$0$$\frac{ 1}{ 2}$
$1$$1$$1$$0$$\frac{ 3}{ 2}$
$1$$1$$1$$0$$\frac{ 3}{ 2}$
$2$$2$$2$$0$$\frac{ 5}{ 2}$

#### Note:

This is operator $\tilde{C_17}$

4

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 ...

5

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 ...

6

New Number: 5.96 |  AESZ: 339  |  Superseeker: 12 28  |  Hash: 41593acc689cf76c174442db98218947

Degree: 5

$\theta^4-2^{2} x\left(10\theta^4+50\theta^3+39\theta^2+14\theta+2\right)+2^{4} x^{2}\left(177\theta^4+1158\theta^3+2007\theta^2+1158\theta+230\right)+2^{8} x^{3}\left(539\theta^4+1344\theta^3-300\theta^2-1068\theta-340\right)+2^{10} 5 x^{4}(2\theta+1)(4\theta^3-642\theta^2-1002\theta-385)-2^{13} 3 5^{2} x^{5}(2\theta+1)(3\theta+2)(3\theta+4)(2\theta+3)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 8, 0, -6400, -249200, ...
--> OEIS
Normalized instanton numbers (n0=1): 12, -339/2, 28, 27639/2, 634692, ... ; Common denominator:...

#### Discriminant

$-(55296z^3-5632z^2+80z-1)(1+20z)^2$

#### Local exponents

$-\frac{ 1}{ 20}$$0$ ≈$0.007072-0.012497I$ ≈$0.007072+0.012497I$ ≈$0.087707$$\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 "5.96" from ...

7

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 $\Q(\sqrt{?})$.

8

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 $Q(\sqrt{?})$.

9

New Number: 8.7 |  AESZ: 106  |  Superseeker: 12 356  |  Hash: fe1c90929d18b81637eaaa93366409ed

Degree: 8

$\theta^4-2^{2} x(3\theta^2+3\theta+1)(11\theta^2+11\theta+3)+2^{4} x^{2}\left(241\theta^4+940\theta^3+1303\theta^2+726\theta+145\right)-2^{7} x^{3}\left(33\theta^4-198\theta^3-607\theta^2-456\theta-117\right)+2^{10} x^{4}\left(239\theta^4+478\theta^3-322\theta^2-561\theta-169\right)+2^{12} x^{5}\left(33\theta^4+330\theta^3+185\theta^2-32\theta-37\right)+2^{14} x^{6}\left(241\theta^4+24\theta^3-71\theta^2+24\theta+23\right)+2^{17} x^{7}(3\theta^2+3\theta+1)(11\theta^2+11\theta+3)+2^{20} x^{8}\left((\theta+1)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 12, 380, 16464, 845676, ...
--> OEIS
Normalized instanton numbers (n0=1): 12, 20, 356, 34561/4, 161840, ... ; Common denominator:...

#### Discriminant

$(64z^2+88z-1)(16z^2+44z-1)(1+32z^2)^2$

#### Local exponents

$-\frac{ 11}{ 8}-\frac{ 5}{ 8}\sqrt{ 5}$$-\frac{ 11}{ 16}-\frac{ 5}{ 16}\sqrt{ 5}$$0-\frac{ 1}{ 8}\sqrt{ 2}I$$0$$0+\frac{ 1}{ 8}\sqrt{ 2}I$$-\frac{ 11}{ 16}+\frac{ 5}{ 16}\sqrt{ 5}$$-\frac{ 11}{ 8}+\frac{ 5}{ 8}\sqrt{ 5}$$\infty$
$0$$0$$0$$0$$0$$0$$0$$1$
$1$$1$$1$$0$$1$$1$$1$$1$
$1$$1$$3$$0$$3$$1$$1$$1$
$2$$2$$4$$0$$4$$2$$2$$1$

#### Note:

Hadamard product $b\ast d$. 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
$Q(\sqrt{?})$.

10

New Number: 16.8 |  AESZ:  |  Superseeker: 12 13064/9  |  Hash: 6df7a0657df14d871f878801092f412c

Degree: 16

$2^{4} \theta^4+2^{3} 3 x\left(8\theta^4-92\theta^3-82\theta^2-36\theta-7\right)-2^{2} 3^{2} x^{2}\left(1088\theta^4-1192\theta^3-4206\theta^2-3004\theta-959\right)+2^{2} 3^{3} x^{3}\left(4560\theta^4+33768\theta^3-25736\theta^2-31794\theta-14623\right)+3^{5} x^{4}\left(100032\theta^4-536736\theta^3-96744\theta^2+29064\theta+61715\right)-2^{3} 3^{6} x^{5}\left(169160\theta^4-91804\theta^3+41266\theta^2+53386\theta+40577\right)+2 3^{7} x^{6}\left(3414784\theta^4+6784176\theta^3+13515220\theta^2+15009792\theta+7398205\right)-2^{2} 3^{8} x^{7}\left(1392976\theta^4+4030952\theta^3+11019392\theta^2+21953818\theta+17966595\right)-3^{10} x^{8}\left(33964576\theta^4+198076768\theta^3+508864632\theta^2+625406200\theta+298971681\right)+2^{3} 3^{12} x^{9}\left(5514696\theta^4+37470916\theta^3+113533894\theta^2+174129464\theta+110762081\right)-2^{3} 3^{12} x^{10}\left(2307232\theta^4-10413916\theta^3-29280133\theta^2+104107498\theta+229795002\right)-2^{2} 3^{14} x^{11}\left(55805104\theta^4+602312152\theta^3+2446466552\theta^2+4496900138\theta+3199953147\right)+3^{15} x^{12}\left(354369472\theta^4+4305262368\theta^3+20428056776\theta^2+44627766264\theta+37585137717\right)-2^{3} 3^{16} x^{13}\left(20975576\theta^4+285973420\theta^3+1472576206\theta^2+3384942194\theta+2924420331\right)+2 3^{18} x^{14}\left(1698304\theta^4+22123696\theta^3+107456180\theta^2+230344688\theta+183698835\right)-2^{2} 3^{20} x^{15}\left(4976\theta^4+61944\theta^3+290160\theta^2+606438\theta+477397\right)+3^{22} x^{16}\left((2\theta+7)^4\right)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, 21/2, 567/8, 2205/16, -261333/128, ...
--> OEIS
Normalized instanton numbers (n0=1): 12, -219/2, 13064/9, -51069/2, 518412, ... ; Common denominator:...

#### Discriminant



No data for singularities

#### Note:

This is operator "16.8" from ...

11

New Number: 21.4 |  AESZ:  |  Superseeker: 12 -73270936/9  |  Hash: ea0a19e2bbcb75ab0103e861337b3f2a

Degree: 21

$\theta^4+3 x\left(573\theta^4+346\theta^3+412\theta^2+239\theta+54\right)+3^{2} x^{2}\left(156063\theta^4+190252\theta^3+259127\theta^2+180550\theta+52656\right)+3^{3} x^{3}\left(26873063\theta^4+49571770\theta^3+76212373\theta^2+6106354\theta+20777568\right)+2^{2} 3^{4} x^{4}\left(820581957\theta^4+2034754740\theta^3+3492838265\theta^2+3133962118\theta+1190460168\right)+2^{4} 3^{6} x^{5}\left(6300362534\theta^4+19676778467\theta^3+37370330484\theta^2+36883703215\theta+15276339186\right)+2^{3} 3^{7} x^{6}\left(909135259983\theta^4+3431279279032\theta^3+7154070873743\theta^2+7670002845554\theta+3413524335720\right)+2^{6} 3^{8} x^{7}\left(6578661109677\theta^4+29157693414323\theta^3+66290838226280\theta^2+76488360260036\theta+36219329073882\right)+2^{5} 3^{9} x^{8}\left(621118841654771\theta^4+3165334819559244\theta^3+7801391210481967\theta^2+9618732171727698\theta+4811767929521208\right)-2^{8} 3^{11} x^{9}\left(1006971762683339\theta^4+5805767585264362\theta^3+154317312352859336\theta^2+20216923376831949\theta+10626549650623836\right)+2^{7} 3^{12} x^{10}\left(65010659394650453\theta^4+418646275727401384\theta^3+1194551569101777997\theta^2+1655275652516026742\theta+910294388309482488\right)+2^{9} 3^{13} x^{11}\left(436618025486403133\theta^4+3107758129098031614\theta^3+9480179119729435361\theta^2+13841832366370196760\theta+7936511785428285000\right)+2^{9} 3^{15} x^{12}\left(3253974365419639343\theta^4+25378994322658850220\theta^3+82459599652886212555\theta^2+1264452814346649241666\theta+7537566452320106004\right)+2^{12} 3^{16} x^{13}\left(7545567178315476151\theta^4+64015361780620462923\theta^3+220793041913868320297\theta^2+354629201172868320135\theta+219217347256983574500\right)+2^{11} 3^{17} x^{14}\left(2307919988301912384493\theta^4+2116501102196476284048\theta^3+7725401378678907468469\theta^2+12964454588967176832582\theta+8293341868591288772952\right)+2^{14} 3^{19} x^{15}\left(119959654773054384185\theta^4+1182698869985222762683\theta^3+4555730636505118858198\theta^2+7970714706688080962144\theta+5266713350723171529432\right)+2^{13} 3^{21} x^{16}\left(800735133480737687025\theta^4+8446905845515949546700\theta^3+34248485289203789137933\theta^2+62351252711799696042686\theta+42485364821351640163944\right)+2^{17} 3^{23} x^{17}\left(130854593635281335447\theta^4+1470740687678042633052\theta^3+6261882273616975205334\theta^2+11841616915017568640538\theta+8308392497254055969184\right)+2^{15} 3^{25} 23 43 x^{18}(\theta+2)(1044375511951501991\theta^3+10371017408154621546\theta^2+34841428344375119043\theta+39324906832335439780)+2^{16} 3^{29} 5 13 23^{2} 43 x^{19}(\theta+2)(\theta+3)(54356917074933\theta^2+414255137487765\theta+802667272851940)+2^{17} 3^{34} 5^{2} 13^{2} 23^{3} 43^{2} x^{20}(\theta+2)(\theta+3)(\theta+4)(13874987\theta+59815761)+2^{20} 3^{37} 5^{3} 13^{3} 23^{4} 43^{2} 163 x^{21}(\theta+2)(\theta+3)(\theta+4)(\theta+5)$

Maple   LaTex

Coefficients of the holomorphic solution: 1, -162, 19710, -2134872, -11551097421/16, ...
--> OEIS
Normalized instanton numbers (n0=1): 12, -219/2, -73270936/9, 864369416211/512, -257294880950199/1000, ... ; Common denominator:...

#### Discriminant

$1+1719z+1404567z^2+725572701z^3+265868554068z^4+73487428596576z^5+15906230508662568z^6+2762406114597811008z^7+391215429129307442976z^8-45665798872080756181248z^9+4422314219437111218274944z^10+356408402046750055533547008z^11+23905779481492695503999027712z^12+1330429645257537748395011567616z^13+610396494807586935900456329951232z^14+2284330751383039846541552834887680z^15+68615965231184830490826918224486400z^16+1614684803424676318995060487706247168z^17+28677036868562929135397434363145650176z^18+361483616787491842925284831575581982720z^19+2882785335303214648482273607706807500800z^20+10936094830042478173776091777549860864000z^21$

No data for singularities

#### Note:

This is operator "21.4" from ...