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ex.24.7.1.31_63_95.a

Base Field
\(F = \) 2.1.2.3a1.4 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 3\cdot 2 )\) 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
\( v(N) = 7\)
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 \(x^{2} - 211275100038038233582783867563b^{3} x + (-211275100038038233582783867563\mu_3 - 211275100038038233582783867563)b \)

Underlying Character

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

Order
4
Conductor exponent
11
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 11 })^\times/U_{\mathfrak{p}^{ 11 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 1 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} - 2a\cdot \mu_3)c + 4\mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left((2a\cdot \mu_3 + (2a + 2))b\cdot c + 4\mu_3b^{2} + (4\mu_3 + (2a + 4))b - 3a\cdot \mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((3\mu_3 + 3)b + 2)c + (3a\cdot \mu_3 + 3a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 - 1)b^{2} - 2b + 4\mu_3)c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 1 } \\ \chi^A\left((-3b + 2\mu_3)c + a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + (a + 2))b + ((a - 2)\mu_3 + 4))c + (3a - 1)b + (a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((3a - 2)\mu_3 + 4)b^{2} + (2a - 2)b + (a\cdot \mu_3 + 3a))\cdot c + (-\mu_3 + (2a + 4))b^{2} + (3\mu_3 + (2a - 1))b + (-a - 3)\mu_3 - 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left((2b^{2} - 2\mu_3b - 2a\cdot \mu_3 - 2a)\cdot c + 3b^{2} + (2a - 3)\mu_3b + (2a + 1)\mu_3 + 2a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + 4)b^{2} + (2a + 4)b + 4a\cdot \mu_3)c + (2a + 2)\mu_3b^{2} + (-3\mu_3 + 4)b + (2a - 2)\mu_3 + 2a - 1 \right) &= i^{ 2 } \\ \chi^A\left((((a + 4)\mu_3 + (3a + 2))b^{2} + (-2\mu_3 + (3a + 2))b - 2\mu_3 + 4a)\cdot c + (2\mu_3 - 2)b^{2} - 3\mu_3b - 2\mu_3 + 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left((\mu_3b^{2} + (2\mu_3 + 2)b + 4)c + 2a\cdot \mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + 2a)\cdot b^{2} + 2\mu_3)c + (4\mu_3 + 4)b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -124589267052306977044889693952a - 360784016118129134629243161472 x^{47} + -603919158888998298761935413744a - 404039529912663422849799374960 x^{46} + (-127765074222583417641594143888a + 14501147787103454438150937920 )x^{45} + -129155484736151248572859709420a - 292733370398154206714357137120 x^{44} + (-298559523614699138856092222520a + 328418929295398831466429571816 )x^{43} + 28246452962134318751968138368a - 357580881064505740264361828284 x^{42} + (-47914567273535937218496773464a + 593696994350662035033737311456 )x^{41} + 307745742273126688189691308748a - 542971808486447219987720628200 x^{40} + (61184314741245270541488174792a + 85462031451304782250357201568 )x^{39} + -136826904250626208796491373248a - 87553978661017786729675602544 x^{38} + (-87437351886799462803079798472a + 621537236420450031720077232880 )x^{37} + -265501388152413043576652521848a - 291316105803544577590474285616 x^{36} + 512842565632613757410106944496a - 405834973688765454040678571920 x^{35} + 480216253033952832212197060008a - 522294080471391287987118537728 x^{34} + -318900347462445643714793698704a - 266099981590038235661279018592 x^{33} + (-412745854007449112856846496516a + 618857474119425554676924151720 )x^{32} + (226637066364158094205053820384a + 220627716911850873123314279360 )x^{31} + 414372427451629763277683180304a - 519184434124907235677563774760 x^{30} + (-583999175661754179410816023288a + 151610964478081947677970365104 )x^{29} + (30843881363297896590243028252a + 477942434158777701534331615004 )x^{28} + -443726843578579772689790956768a - 604559936721421655465964868824 x^{27} + 212609887534195199912675009216a - 354560174916528403589422616576 x^{26} + 190728928838596200310316865648a - 20578912887632320184229139824 x^{25} + -236813672684665525908980458966a - 257934769491994815171423812288 x^{24} + (-307251213975128383458517092608a + 292736455146846869513747428128 )x^{23} + (99849705527856340052684137288a + 595908341088404890033157738960 )x^{22} + 116555282900877242998102682256a - 301687233729891336547474695648 x^{21} + -226483141201753607578940993480a - 411779745335860640810881775752 x^{20} + 414675652943706133804025001000a - 578349478250850057883896322192 x^{19} + 225124342412058416757138846536a - 7382260130137119702465222368 x^{18} + 527510694314902240957522853024a - 400765937721728943758604472080 x^{17} + 327149085693970045210124585432a - 218219060512023659559802605768 x^{16} + (37241200105064276578972497664a + 411372564506943711283829763600 )x^{15} + -177330176172526947292314896008a - 621032260315324107846321972624 x^{14} + -83283634639980413958335229856a - 602096845976443933818080674816 x^{13} + (471370529806618083159137138592a + 382389435142169944151955506028 )x^{12} + (-162611144455225692824013555104a + 106896605951457804325779843424 )x^{11} + 132961954364148392369220057224a - 603696430169363468060174399608 x^{10} + 246413364283876006639751727808a - 85243956970403694345606485424 x^{9} + -452492242931607729738496668456a - 232733905123759253333325389560 x^{8} + -69152926779748584796063339648a - 56970859895579061118088924128 x^{7} + (-211521863731598009657010067864a + 562647404332911667156324131376 )x^{6} + -10920492945741279419008622240a - 309021676723930460486733903488 x^{5} + -137709544202390362829838674404a - 358562536117244216528726811768 x^{4} + 582513220107638496248488774528a - 429768084362255297180362614592 x^{3} + -467257591333233272250791422768a - 567489047084562511322628700608 x^{2} + 248775534827495036047671175912a - 319295259736163192761723941040 x - 81777004136680948910310860992a + 498238901112531836766370952266 \)
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