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ex.24.10.1.127_255_383.d

Base Field
\(F = \) 2.1.2.3a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 )\) 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 \(x^{2} - b^{4} x + b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -351750906685552534296702778256a - 589153484449285808298340694984 x^{47} + (55149946770643039941178364588a + 450955358421375495541435692208 )x^{46} + -459017496688376080595489204500a - 313993239468692426593775686176 x^{45} + -305140711199783901487201953988a - 241379674200296642780425699012 x^{44} + (-14336800364884510088452962192a + 320511671750062561127260150608 )x^{43} + (-263744996013371572847068972028a + 526919721502398867809616135020 )x^{42} + -308129274787794047338730292376a - 342021479991111727173535391928 x^{41} + 619701229945062286216200306024a - 73706378125701585265570310116 x^{40} + -17512518101806663207154898056a - 389982874646416031914139036976 x^{39} + (-488456933968106796433986055412a + 587721924223307331101834602992 )x^{38} + (206281846401954795373646277316a + 211477080176811446204461252448 )x^{37} + (-162261445088034809539450666504a + 200225814519077106060107262412 )x^{36} + (-105226327535401433037091563992a + 345538838368671981277201743856 )x^{35} + -162887445995441448444441583108a - 350221628533223193613922112464 x^{34} + (356922633435006881161522143672a + 584614885206745441564949435480 )x^{33} + 163561805726351332768848517632a - 53341116511574408294195785168 x^{32} + (-403541815211701880800633858720a + 523580814971957858512522228608 )x^{31} + (387049567455240563541180133864a + 496630319898448279987138474080 )x^{30} + (59832433303061816888191074344a + 589299856205442955174200953208 )x^{29} + (571081968417318429963360393544a + 172337570664430306454695653104 )x^{28} + (628632072392469510252231510080a + 55714402252135540102163494016 )x^{27} + 425292226516193043206492706356a - 152088385031313992120246648440 x^{26} + (-377068721383102824758778370544a + 27607353942846755697484844704 )x^{25} + -208223121684042061175087180034a - 223482549206652852801401335064 x^{24} + (113792635683440022746752506792a + 401157142617893988021790193632 )x^{23} + (30767600864154410859229460600a + 218815062350703577861897859064 )x^{22} + (-159923830199129051118228316032a + 421363635615550230534777985560 )x^{21} + (279142329497881634252002760820a + 296677363604799985286507162200 )x^{20} + -329725102478986113574712748896a - 225654192910132032257113747376 x^{19} + 8661545528229211206332486444a - 56230208164599608916345625064 x^{18} + -292015644099576714281356257176a - 135393926002827714328667855104 x^{17} + 501534780584596511576852072444a - 123152949522068226084787832112 x^{16} + (-138791830175966869962002270768a + 570242745444959381035647090688 )x^{15} + -599446619210159609349187027752a - 87432075332874830278790725736 x^{14} + (-412576368703253390557129213232a + 266589788093749276010614310760 )x^{13} + (305579600064346144502903261728a + 329949360489277546107411827608 )x^{12} + (-485677568861244095417895646880a + 191204431745767861929732475536 )x^{11} + (418327168594690797567108958872a + 327893886867668550212769912472 )x^{10} + 536581798470351692934193612792a - 104069485265767398787919981136 x^{9} + 463790191664172864606938378432a - 84865296705266668044094297552 x^{8} + (-120016239578177001663652344384a + 214705922262010822558861734736 )x^{7} + -373274024734307698331637617792a - 84795658396353435824227773952 x^{6} + 111452869246912389012336335864a - 442317732934410337857643516496 x^{5} + (-471349764388338307608316366144a + 167161329695058601005281841712 )x^{4} + (534383997964270650211105263632a + 362306830285469930236036611296 )x^{3} + 402875918955408400551212435304a - 142015869933171069483583684920 x^{2} + (444632555541142791360501711824a + 466062277447061562690880036000 )x + 546150093423241578357968886656a + 203649695270571920098253017542 \)
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