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ex.24.10.1.131_259_387.a

Base Field
\(F = \) 2.1.2.3a1.3 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 7\cdot 2 )\) 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 \(x^{2} + 90546471444873528678335943241b^{4} x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -248992593236049286289552012336a - 547095474376502498932031775608 x^{47} + 378285599511656660901679841828a - 600543819938515780745227327004 x^{46} + 614484970678849249959970025620a - 318846928403249210876650358768 x^{45} + (419675180850022012198694278084a + 187201756008502001794135267648 )x^{44} + (400092328214953337257097891240a + 539291897187493714297452499248 )x^{43} + 571849226858323307047920722388a - 471958424668020740830190981028 x^{42} + 142970202863851689045056464824a - 520022575124522047173690699816 x^{41} + 556091698616587395605066693240a - 332630598794016870218453996132 x^{40} + (-168543507842370301557891613304a + 112094893853826826252048861840 )x^{39} + -80495886656888616087937705944a - 544811501871879739828141517552 x^{38} + (139945459352020653245227127004a + 531911662855453523620786327136 )x^{37} + (148583435929438400809978257880a + 403948562031106437767096229204 )x^{36} + 225519022016157924948913578352a - 249592008642385007119799277184 x^{35} + (-406212594409847168491326370364a + 26783932998609587357451143728 )x^{34} + 240239500245024383200119392456a - 473785417648922786477385754288 x^{33} + -112893332196950197300778549756a - 611820171995169003216179417584 x^{32} + (526536850174003630964010931776a + 572241324510816373478058283040 )x^{31} + (-598054982493017527449784490264a + 565489030203052182489718374880 )x^{30} + -546820003925803864234297671632a - 320249242500043279816561350944 x^{29} + (-264891168498747434697604031692a + 90011142672883961359541630032 )x^{28} + 495829089816449641788844151776a - 289502076838664135673793455568 x^{27} + (327991399410092179769715097656a + 479797300845919750582563126568 )x^{26} + (456171759431758259842643206120a + 436308044384555915152274138752 )x^{25} + (135621569329342304768922125778a + 364374073520815669615739958988 )x^{24} + -64285179638125557183629131080a - 418339746383493548481138831792 x^{23} + (-221109768300775574391795499052a + 196924343842037213135337465408 )x^{22} + (567239224534392628203107837608a + 259694926737763719393836521160 )x^{21} + 104719305579091156343934121368a - 409824852438904050565908197480 x^{20} + 491628244484214269083691076720a - 584143998587227567125985608912 x^{19} + -68646385441315584253041310852a - 252719291840559113512665990352 x^{18} + (-536772338460828939912241592280a + 293038237606971559318108026304 )x^{17} + (244241220265866030447408250412a + 418631189914955360445908830504 )x^{16} + -54869611717458016587088545744a - 392893698079042603081817872480 x^{15} + -182022313833686532748497446080a - 387856965323678038896700531984 x^{14} + -192674485685500309217629417848a - 163032406359155306853627031720 x^{13} + (85270214432871695673519022680a + 629965186445977848766424877416 )x^{12} + -50519619833974934286497580560a - 267651580502254047122121472000 x^{11} + (172475926024156671845704160544a + 502009708201041655330357980632 )x^{10} + 582428939428855482991262213104a - 473983527795884318471712879408 x^{9} + (560864061613612218406166459064a + 14990830183868132927014410360 )x^{8} + -538937416660875912512571455552a - 587637770667897976781421925776 x^{7} + (310107703578748338100383416160a + 281667906104030407768657495200 )x^{6} + (195794134848649076848242966640a + 630589733758473273541452721888 )x^{5} + (76064800638549087532814323672a + 494493749950124495491865233864 )x^{4} + (466570567448124135752365426384a + 385356451972668045002557033280 )x^{3} + (93510333634426356713581345496a + 545649521672166618233008256624 )x^{2} + -290429437007597225058257492896a - 101254708369759863602490201360 x - 120291009387149932969444840540a + 43913759865665532042398395678 \)
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