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ex.24.10.1.31_63_95.b

Base Field
\(F = \) 2.1.2.3a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 5\cdot 2 )\) View on LMFDB ↗
Description
exceptional, SL(2,3)
Construction
Extension to \(I_F\) of \( \tau = \operatorname{Ind}^{I_K}_{I_L} \chi \), with \(L, K\) and \(\chi\) as below
Semistability defect
\( e = 24\)
Conductor exponent
\( v(N) = 10\)
Character Order
4

Triply Field

The inertial type \(\tau\) becomes an induction (triply imprimitive) over: \( L = F(\mu_3, b) \), with \(\mu_3\) a root of \(x^2+x+1\) and \(b\) a root of \(x^{3} + a \)

Inducing Field

The inertial type \(\tau\) becomes reducible over \(K = L(c)\), \(c\) a root of \(x^{2} + 253530120045645880299340641075b^{3} x + (253530120045645880299340641075\mu_3 + 253530120045645880299340641075)b \)

Underlying Character

Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:

Order
4
Conductor exponent
20
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 20 })^\times/U_{\mathfrak{p}^{ 20 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 1 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} - 2a\cdot \mu_3)c + 4\mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 4)\mu_3 + (2a + 2))b\cdot c + 4\mu_3b^{2} + (4\mu_3 + (2a + 4))b + (-3a + 2)\mu_3 - a - 3 \right) &= i^{ 0 } \\ \chi^A\left((3b - 2\mu_3)c + a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((\mu_3b^{2} + (2\mu_3 + 2)b + 4)c + 2a\cdot \mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + (2a - 2))b^{2} + (\mu_3 + (2a - 1))b + ((-3a + 4)\mu_3 - 2a - 1))c + ((a - 2)\mu_3 + (a + 2))b^{2} + ((2a - 1)\mu_3 + (3a - 3))b + (-2a - 2)\mu_3 + a - 3 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 4)\mu_3b^{2} + 2a\cdot b + (4a\cdot \mu_3 + 4a))\cdot c + ((2a + 2)\mu_3 + 4)b^{2} + (-\mu_3 + 4)b + (-2a + 2)\mu_3 + 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left((((3a + 4)\mu_3 + (3a + 2))b^{2} + (3a\cdot \mu_3 + (a - 2))b + ((2a - 2)\mu_3 - a + 2))c + 4b^{2} + (2a - 3)\mu_3b + (-2a - 2)\mu_3 + 2a - 3 \right) &= i^{ 2 } \\ \chi^A\left((-2b^{2} - 2\mu_3b - 2a\cdot \mu_3 - 2a)\cdot c + 3b^{2} + (2a + 3)\mu_3b + (-2a - 3)\mu_3 - 2a - 3 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + 2a)\cdot b^{2} + (3a + 2)b + (-3a + 2)\mu_3)c + (3a - 3)b + (-a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 4)\mu_3 + (a + 2))b^{2} + (2\mu_3 + (3a - 2))b + 2\mu_3)c + (2\mu_3 - 2)b^{2} - \mu_3b + 2\mu_3 + 2a - 1 \right) &= i^{ 2 } \\ \chi^A\left(((-3\mu_3 - 3)b - 2)c + (3a\cdot \mu_3 + 3a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + 2a)\cdot b^{2} - 2\mu_3)c + (4\mu_3 + 4)b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 - 1)b^{2} - 2b + 4\mu_3)c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 3 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (204877173336531253610404736268a + 534322942957277372268937771760 )x^{47} + (102711657842312674434152758636a + 338309939558138640236249524120 )x^{46} + -443638256367908579997893398192a - 396706925713637381522345884344 x^{45} + 367340012954621354173123266468a - 274937321820648233620339568392 x^{44} + 21134914149352424957913193352a - 185859390076785179458015834452 x^{43} + (348893683114590120788306581880a + 562072354227509807617512450392 )x^{42} + -427621297928706847704071131176a - 215612908881534063290356826072 x^{41} + -599232966760779965909563990504a - 281464412881980787275173113904 x^{40} + 40365291195998274538867536984a - 366765246800898265779653377232 x^{39} + (-515633639071537499820241872232a + 104990648305490733176179250436 )x^{38} + (-1058877159678794508706831548a + 14867939983257372106162349432 )x^{37} + (497526860431080946868097121504a + 148496492997903421954655693672 )x^{36} + (-50082816931958533637695557288a + 388226765641065174273930878816 )x^{35} + (584817596311241897910428393792a + 145587919080967546064499266800 )x^{34} + -549284872033528412866486599784a - 541630644936662961218464152328 x^{33} + -597587148926966608636325041816a - 538236098669244335886768147400 x^{32} + (-98037806108066812508163081840a + 60282123693551311260139171296 )x^{31} + (14592277691301470103595744240a + 234119519204551279193981430668 )x^{30} + -118872764674198098697739844804a - 398825722959097832128947470848 x^{29} + (593747432460128799132633912944a + 67419953808592274005963133240 )x^{28} + (161804621275934167352506492832a + 568367911775030098358294972496 )x^{27} + -611861125945575013765414297304a - 23446011637674226308785530592 x^{26} + (-463353053317562383300380615296a + 264118538258452675761506728656 )x^{25} + (187640004214864442673936006958a + 370280088810515851135449645664 )x^{24} + 400358420811516242511878333440a - 238405127129199924189378458152 x^{23} + 508348344513987682533689123432a - 523114159108782653224253149352 x^{22} + -607261481686501164393571151336a - 214563047224907954041150332896 x^{21} + (-115936766390837836827583797680a + 415791189533618540864604658608 )x^{20} + (417783818100579817219548712924a + 75019237723006122419442824576 )x^{19} + (593422947861648428261940160680a + 301180511668345522885905178736 )x^{18} + -515989052897642374134172827944a - 35504852262753161063749758720 x^{17} + (-117617687734554792968780706944a + 230345054026944819293906417056 )x^{16} + (496315140035848300730687846336a + 352495130100737643336710467488 )x^{15} + -2333486314907892822828032436a - 210555706151829777914263300832 x^{14} + -389525885397715612933275268472a - 22298989724928687009958161160 x^{13} + -62363955118687931648243642992a - 399246047308968287814391240348 x^{12} + 425802035375302169274415799624a - 567521125201910778848264846480 x^{11} + (-313098656394711111225631375520a + 567184003412630161064226872448 )x^{10} + (-156430297136372024228885700664a + 611874145153789590718056439120 )x^{9} + -391642299928242010750817058712a - 512033821886636298927980488160 x^{8} + (60387911212261931122195594864a + 2374535317110473207955739472 )x^{7} + (617332540961128125436702043708a + 399774938136009290254165255328 )x^{6} + -576301049715880301581401167264a - 307883407068243771024800558248 x^{5} + (-584080124193223869542456158888a + 303327505606227135201817575472 )x^{4} + 396038535352873268332659390272a - 434240666585321521631944301376 x^{3} + (-227572233064301103866584479072a + 304147465687250809731235185056 )x^{2} + (269086833475125812908043533800a + 452774315506142855125682576928 )x + 628489024658343294800097457264a + 433869224616606860673305485814 \)
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