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ex.24.10.1.33_67_101.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 \(x^{2} + 253530120045645880299340641075b^{3} x + (63382530011411470074835160269a\cdot b + 253530120045645880299340641075)b \)

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^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 - 2)b^{2} + 2\mu_3b + 4)c + ((2a + 4)\mu_3 + (2a + 2))b^{2} + ((2a + 3)\mu_3 + (2a + 3))b + 4\mu_3 + 4a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 4)\mu_3b^{2} + (4a - 2)\mu_3)c + 4\mu_3b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left((((a + 2)\mu_3 + (2a - 1))b^{2} + (a - 1)\mu_3b + ((-2a - 2)\mu_3 + (4a + 2)))c + (3\mu_3 + (3a - 2))b^{2} + (\mu_3 + (a + 1))b + (2a + 2)\mu_3 + 3a + 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 1)b^{2} + (2a + 2)b)\cdot c + (2a + 4)b^{2} + 2a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((3a - 1)\mu_3 + (3a - 2))b^{2} + (3a + 3)\mu_3b + ((2a - 3)\mu_3 - 3a + 4))c + ((2a + 3)\mu_3 + (3a + 4))b^{2} + ((2a + 3)\mu_3 + 3a)\cdot b + 3\mu_3 + 4 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + (2a - 2))b^{2} + ((a + 4)\mu_3 + 3a)\cdot b + ((-2a + 4)\mu_3 - a + 4))c + ((a + 4)\mu_3 + (3a + 2))b^{2} + (\mu_3 + 1)b + (2a + 2)\mu_3 + 2a - 3 \right) &= i^{ 2 } \\ \chi^A\left(((3a\cdot \mu_3 + 4)b^{2} + (-2\mu_3 + (3a - 2))b + ((-a + 4)\mu_3 + (2a - 2)))c + ((3a + 2)\mu_3 + (2a + 4))b^{2} + ((3a + 1)\mu_3 + 4)b + (3a + 3)\mu_3 + 4a \right) &= i^{ 0 } \\ \chi^A\left((((a + 4)\mu_3 + (2a + 2))b^{2} + (2a\cdot \mu_3 + 4)b + (2a\cdot \mu_3 + 4a))\cdot c + (2\mu_3 + 2a)\cdot b^{2} + (-2\mu_3 + 2)b + (3a - 2)\mu_3 + a - 3 \right) &= i^{ 2 } \\ \chi^A\left((2\mu_3b^{2} + ((3a - 2)\mu_3 + 4)b + ((a - 2)\mu_3 + (4a + 4)))c + (3a\cdot \mu_3 + 4)b^{2} + ((3a + 1)\mu_3 + 4)b + (-3a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + (2a + 4))b^{2} + ((a + 4)\mu_3 + (2a + 4))b + 4)c + (a - 3)\mu_3b^{2} + ((a - 1)\mu_3 + (3a + 3))b + (4a + 4)\mu_3 - a + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 2)\mu_3 + (3a + 2))b^{2} + ((a + 4)\mu_3 + (2a - 3))b + (2a\cdot \mu_3 - a + 4))c + (2a - 1)\mu_3b^{2} + (a\cdot \mu_3 + 2)b + (4a - 2)\mu_3 + 4a - 3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 4)\mu_3 + (3a + 4))b + ((a - 2)\mu_3 + (a - 2)))c + ((a - 1)\mu_3 + (a - 1))b^{2} + (-\mu_3 - 1)b + (4a - 1)\mu_3 + 4a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)\mu_3b^{2} + (2a + 2)\mu_3b)\cdot c + (2a + 4)\mu_3b^{2} + 2a\cdot \mu_3b + 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} + 382452038323262919676283539244a - 598674030755459596329358279672 x^{47} + (-391895837861092639857986869748a + 381449457550605827933263459740 )x^{46} + (-497366064280681698363751231080a + 410853852483517207689921541888 )x^{45} + -79878105275693089133265798520a - 19940714578234327928638576796 x^{44} + -599929823678725867038059919860a - 207996751599543196730267231828 x^{43} + -48360670393068555248104103144a - 166301206650915338225115619464 x^{42} + 280962653370320912256751667252a - 388423398892006157486190084328 x^{41} + (184086242747652929300585873788a + 181167019746660702205563499312 )x^{40} + (-365123875511397809957221161296a + 121612259722216721218530379992 )x^{39} + (337830378178375373735765941372a + 30781533581427130012248785484 )x^{38} + -254951078581147921848591668684a - 89160388791032618589228812492 x^{37} + (277630560368363580190897079608a + 294712560548445252449866131016 )x^{36} + (-273417991971005117011702085600a + 247621082011164301446132260488 )x^{35} + (-463862838914383234607620931176a + 219765125169185923972565263488 )x^{34} + (600441301248192839199028691220a + 162224002041680967657224377048 )x^{33} + 530926597811657741454897042740a - 323187317484998058513052668256 x^{32} + (115036800726492049835774501192a + 271086885327743264452183290208 )x^{31} + (-213593924001995548902015376812a + 388525444330404881248461157524 )x^{30} + 327544678915832988321734918128a - 420124648426471347886329408312 x^{29} + (225415304901043912880143216904a + 550705080296912508202707279288 )x^{28} + -546409850645843250108440631144a - 448882787577916749415203704552 x^{27} + (154994343553694416260193510600a + 227172290656253767241984409052 )x^{26} + 469384284993982534953655868176a - 92722045484936585078040243760 x^{25} + (-46598740071422227281776481286a + 456420013197941018111655249916 )x^{24} + 238611155734886648791729474248a - 257514592351708538162905955464 x^{23} + -212266440178414105871466034852a - 314318232221013970998238804512 x^{22} + (-167472148580567603672935930912a + 621845756451189796320901517728 )x^{21} + -290360513593624509120550614268a - 110268628245050442375386486048 x^{20} + (-378858095918174505184786438188a + 29174276239433613489301318600 )x^{19} + 473074211732637339906566639848a - 328199408956037247374927072800 x^{18} + -172993483467819320611724859400a - 280710327456766130100002573752 x^{17} + 146871805020258564420175615304a - 615564169802873620191930698344 x^{16} + -73832294133476670631061485888a - 168072476482522649531017600288 x^{15} + 471175743304099234980101656156a - 354802985224704571766804914672 x^{14} + (25965929260228461795971442700a + 487680502713026037624319090864 )x^{13} + -545804813933508365435521718184a - 106039432701844144334875616036 x^{12} + (342616308543468076559233868120a + 631367947276207443081594377856 )x^{11} + (-14706748311755621776634571104a + 372546585723626303996688438480 )x^{10} + (395545294601721927468251140448a + 493586536253338263600815877784 )x^{9} + 36600345924495814169465056832a - 546307867434553912336979265856 x^{8} + (40826192829887376848335464976a + 356306091344299403440598529040 )x^{7} + -593721843762555364390362140348a - 501179110907090643868305422512 x^{6} + -109978611658861846982055519256a - 385256255625637266910332282848 x^{5} + -267929546473531432808123245024a - 363826652738392391401496666288 x^{4} + -595302711431287024999962879176a - 529569299188395354771807777472 x^{3} + 224117003419804070015132158852a - 263480154663879421646221075288 x^{2} + 11828136358536510557591328280a - 141262676297819888789710213968 x + 355936227017083306807808115052a + 375467256979033465233630315338 \)
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