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ex.24.10.1.131_259_387.c

Base Field
\(F = \) 2.1.2.2a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 3\cdot 2 )\) 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 \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{4} )x + b \)

Underlying Character

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

Order
4
Conductor exponent
18
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 18 })^\times/U_{\mathfrak{p}^{ 18 } }\) :
\(\begin{array}{l} \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + (3a + 2))b^{2} + (-2\mu_3 - 2)b - 2\mu_3 - 2)c + ((3a - 3)\mu_3 + (3a - 3))b^{2} + ((3a - 3)\mu_3 + (3a - 3))b + (a - 3)\mu_3 + a - 3 \right) &= i^{ 0 } \\ \chi^A\left((4\mu_3b^{2} + (4\mu_3 + (2a + 2))b + (2a + 2)\mu_3)c + ((2a + 2)\mu_3 + 4)b^{2} + ((2a + 3)\mu_3 + 3)b + 4\mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a - 2)b^{2} - b)c + (3a - 2)b^{2} + 1 \right) &= i^{ 0 } \\ \chi^A\left((\mu_3b^{2} + (2a + 4)\mu_3)c + 4\mu_3b + 4\mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left(\mu_3c + 1 \right) &= i^{ 0 } \\ \chi^A\left((b^{2} + (2a + 4))c + 4b - 3 \right) &= i^{ 0 } \\ \chi^A\left(a\cdot c - 2b + 1 \right) &= i^{ 0 } \\ \chi^A\left((a\cdot \mu_3b^{2} + 4\mu_3b + (4\mu_3 + (2a - 2)))c + ((2a + 2)\mu_3 + 4)b^{2} + ((2a + 3)\mu_3 + (2a + 3))b + (2a + 4)\mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left(((4\mu_3 + 2a)\cdot b^{2} + (4\mu_3 + 4)b + 4)c + 4\mu_3b^{2} + (-3\mu_3 + (2a - 3))b + 4\mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left((((a + 2)\mu_3 + (a + 2))b^{2} + ((a - 2)\mu_3 + (a - 2))b - 2\mu_3 - 2)c + ((a - 3)\mu_3 + (a - 3))b^{2} + ((3a + 1)\mu_3 + (3a + 1))b + (a - 3)\mu_3 + a - 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a - 2)\mu_3b^{2} - \mu_3b)\cdot c + (3a - 2)\mu_3b^{2} + 1 \right) &= i^{ 0 } \\ \chi^A\left(((4\mu_3 + (3a + 4))b^{2} + 3a\cdot \mu_3b + (4\mu_3 + 4))c + ((2a + 2)\mu_3 + 4)b^{2} + ((2a + 3)\mu_3 + 3)b + 4\mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 2)\mu_3 + (3a + 4))b^{2} + ((3a + 4)\mu_3 + (3a + 4))b + ((3a - 2)\mu_3 + (2a + 2)))c + ((3a - 3)\mu_3 + 2a)\cdot b^{2} + ((a - 1)\mu_3 + (a - 3))b + 3a - 1 \right) &= i^{ 1 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-324811737210834938235914430008a + 530336601826584742188590253352 )x^{47} + (-622263968423588438681213412380a + 200079408791257440547512269464 )x^{46} + (486083667989806280561784977884a + 365763010793949107251129094816 )x^{45} + (-330957191435870433946505162172a + 393957809184052818923626709396 )x^{44} + (-534772172341331675454979748568a + 163633740730821243674922731824 )x^{43} + (-204095758243420347659890319504a + 165877072115090046663766613516 )x^{42} + -190979092924803520841988080872a - 423525499167173600585755688672 x^{41} + -36964559596571656445871309644a - 178064328400407876197599431828 x^{40} + (-419068616856830893374977713480a + 21275971857492934575186803248 )x^{39} + (-519262688064473918364219736192a + 318382115411667859815581461248 )x^{38} + (-463430670467227142111632403500a + 388614790509747811259159830288 )x^{37} + (-346512037592376718579840433568a + 443342449708151461015596488940 )x^{36} + (270474871068469038765496861216a + 342835576268205984555539178608 )x^{35} + 170525698112316009345076525836a - 569011856184030013901763926200 x^{34} + (-90348820726414813259461325256a + 144339402994281252798986620120 )x^{33} + (100531205307232730309897232404a + 4735725537812089401755944592 )x^{32} + (171434637955739265684313414560a + 393701445022045967264782167968 )x^{31} + (417999428281108136540457669624a + 502862488327516922936074839424 )x^{30} + (276570642870021720433649506848a + 250387705601210329682169467352 )x^{29} + 487167358621248772388327988532a - 226298818008268120622440049792 x^{28} + -293096816189161914939332538416a - 7301811803442387653035659616 x^{27} + -281887553104829702517458528140a - 188667709210949272344260252672 x^{26} + -628925606451248514647104259048a - 539960621251023353152474901144 x^{25} + (-51939315469950619137611894626a + 81110256400190571739187233844 )x^{24} + (244039535298069435836869711112a + 180907879632954387887503160704 )x^{23} + (-3489279976080156313817287416a + 596130496070525179325623214744 )x^{22} + 22876473396223180133227975824a - 409701064624640648242761031512 x^{21} + (-473237101698080898636101009620a + 501535814166090439939516880672 )x^{20} + 205381321872889052878841666480a - 161217039341439490318028572112 x^{19} + (-168340579089814047701883027396a + 533631644873861880636982896520 )x^{18} + (560009922025678237167103356512a + 163137605767424560390173785824 )x^{17} + -557864772443975166803410001140a - 592327542609663387509558790784 x^{16} + (-535204761028126964815691439200a + 9747608597587575975673526016 )x^{15} + (-501884104116775633426943696168a + 586571577222364695449158112896 )x^{14} + 272295064163695950638085672800a - 155854020706340664996815696968 x^{13} + -15299362868070687729513332696a - 436579326814647142027551346616 x^{12} + -30434612323112104027442287520a - 290985945168293814366034652640 x^{11} + -261498474209756036210237777280a - 516729803061758694811619546680 x^{10} + (-78677634640333974469458084776a + 415513868789772011937012496896 )x^{9} + -109972173920381094303807212128a - 277160137962194416862925361448 x^{8} + -510056431498998962320963796016a - 473277302319602284816737569840 x^{7} + -117007110379035831041824139504a - 298763194978299869214914742240 x^{6} + -290538958345840059020040194136a - 381795813273325707315173850992 x^{5} + (419906383322227401282389758400a + 253171600300373864508669791896 )x^{4} + 301562506476070368129180766000a - 12479688407866319894660107200 x^{3} + 250773736965196582357090094176a - 60370464090602595766000544792 x^{2} + (19447356802513536432834881112a + 369236769498963405345903442912 )x - 125379181494438594944263908894a + 553984150117684606724658596598 \)
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