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ex.24.10.1.33_67_101.c

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 \(x^{2} + 90546471444873528678335943241b^{3} x + (-90546471444873528678335943241a\cdot b + 90546471444873528678335943241)b \)

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^{ 2 } \\ \chi^A\left(((2a\cdot \mu_3 + 2)b^{2} + 2\mu_3b + 4)c + (2a\cdot \mu_3 + (2a - 2))b^{2} + ((2a - 3)\mu_3 + (2a - 3))b + 4\mu_3 + 3 \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^{ 0 } \\ \chi^A\left((((2a - 3)\mu_3 + (3a - 1))b^{2} + (2\mu_3 + (2a + 1))b + ((3a + 2)\mu_3 - 3a + 2))c + ((3a - 2)\mu_3 + (a + 4))b^{2} + ((2a + 3)\mu_3 + (a + 1))b + (2a + 4)\mu_3 - 2a + 1 \right) &= i^{ 0 } \\ \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 + 2)\mu_3 + (2a + 3))b^{2} + (a + 1)\mu_3b + ((2a - 2)\mu_3 - 2))c + (3\mu_3 + (a - 2))b^{2} + (3\mu_3 + (a + 3))b + (2a - 2)\mu_3 - 3a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 2)\mu_3 + 2)b^{2} + (a\cdot \mu_3 + a)b + ((-2a + 4)\mu_3 - a + 4))c + (3a\cdot \mu_3 + (a - 2))b^{2} + (-\mu_3 - 1)b + (-2a - 2)\mu_3 - 2a + 1 \right) &= i^{ 2 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} + ((a - 2)\mu_3 + (2a - 2))b + ((4a - 2)\mu_3 + (4a - 2)))c + ((3a + 4)\mu_3 + (3a + 4))b^{2} + (\mu_3 + (2a - 3))b - 2\mu_3 - 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 2)\mu_3 + (2a + 4))b^{2} + ((2a - 2)\mu_3 + (2a + 2))b + (4\mu_3 - 2a - 2))c + ((3a + 1)\mu_3 + 2)b^{2} + ((3a + 1)\mu_3 + (3a + 1))b + 4\mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + 4)b^{2} + ((3a + 2)\mu_3 + 4)b + (-3a - 2)\mu_3)c + a\cdot \mu_3b^{2} + (3a - 1)\mu_3b + (-a + 3)\mu_3 + 4a \right) &= i^{ 0 } \\ \chi^A\left(((4\mu_3 + 2)b^{2} + ((3a + 4)\mu_3 + (3a + 4))b + ((4a + 4)\mu_3 + (a - 2)))c + ((2a + 1)\mu_3 + (3a + 2))b^{2} + ((3a + 3)\mu_3 + (3a + 3))b - 2\mu_3 + 3a - 1 \right) &= i^{ 0 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + 2a)\cdot b^{2} + ((a + 4)\mu_3 + (3a - 2))b + (-3a - 2))c + ((2a - 1)\mu_3 + (3a + 2))b^{2} + ((a + 3)\mu_3 + (3a + 3))b + (-2a - 2)\mu_3 + a - 1 \right) &= i^{ 2 } \\ \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} + -238067837838019993960529701696a - 593480996043203622202417394496 x^{47} + -358808139263465210752257165172a - 66083057297674939819884647500 x^{46} + 201459979617873594674530716532a - 516665427794432642706965488736 x^{45} + -9150250202494153324447552828a - 611751772257001584278107906252 x^{44} + 478382256771694612933340607644a - 321741641074195894872935740252 x^{43} + (613915168497710657594721682732a + 586761159436867370933450040976 )x^{42} + 330053356819587558142924324856a - 513445507516398203085748539400 x^{41} + (212232517034964762290349347196a + 335187680635397038117290367840 )x^{40} + (583158721008632011514411170768a + 561132519346701654315565081952 )x^{39} + (42832990659384155724821823388a + 372857854672464368188117649612 )x^{38} + -70101148816917252394337099968a - 392591275171510212421629569572 x^{37} + (176635972974226801332608555876a + 482101724519996802952532714184 )x^{36} + 552436966299654293538485438168a - 26412957186646396694767675048 x^{35} + -214833601253165985453965437660a - 357308092570881738618722116048 x^{34} + (-418807858023769184635873579252a + 499460778019513111489452591160 )x^{33} + (-604683632806552058208865073908a + 101600730022944740788439313336 )x^{32} + 418567858378928967847982727756a - 585401009465441245072656360672 x^{31} + (271452136655728471562788391436a + 159135680202194816034506017964 )x^{30} + -24639248290945266026752606660a - 125750214664851302729590684864 x^{29} + (237157507345146340185352635128a + 487614644033725228424031366856 )x^{28} + (-539751400437781777381865734952a + 151958264936152339765036557192 )x^{27} + 178829629958371128425424393592a - 620188536812762630534162529788 x^{26} + (-510431816267165176996264086404a + 401210502444290030861176160152 )x^{25} + (532692923428340337851866490210a + 464019138491891171716940259164 )x^{24} + -608652118075784138576563925264a - 407350411284040533649140416656 x^{23} + (-603198538764474513157811993804a + 513468717737007708463012883952 )x^{22} + (178458834235565406266334659320a + 140325075374279694320719098552 )x^{21} + 460907549291806365726891095044a - 195224164972371992657034114584 x^{20} + 43320051661363651866769352004a - 47694147375788096296275008264 x^{19} + 187993970599367042591499890736a - 21585316119335261033386338456 x^{18} + -100538428943107268266016226848a - 468231045335328136606287585232 x^{17} + 242936105727968926085794425168a - 209168588467076468767320490456 x^{16} + 449480609844617207202491266168a - 23578548947615231674061028432 x^{15} + 144996341529291505937383782580a - 110632918524674841978532141216 x^{14} + (-92991110797788949565149776764a + 322855240864031518254897615464 )x^{13} + (-385004631095714699265571986424a + 198493958830816953483524635284 )x^{12} + 456569790871196133470026400320a - 497506353798426201250321067904 x^{11} + (585176276235929855255260026696a + 232254513135085028695853598072 )x^{10} + (343943119566669231108105582880a + 15460749882383334010524299832 )x^{9} + (-140799802314299774970796937688a + 467534114841217248642493219568 )x^{8} + 271940127284679266198233818080a - 611216181388195076397722696168 x^{7} + 319585243310087998581347044060a - 2656539528186988782706472352 x^{6} + (-632324696380636937360316694776a + 10002540154561704543965217608 )x^{5} + 467843503429803260114878216720a - 392133171638311328252726510480 x^{4} + 422157381457784876497323458904a - 517081648263149975872284132848 x^{3} + 149865333082659491625133317660a - 549046073849107669472567553784 x^{2} + (285100929568418833696775817784a + 403829709266279857175346283592 )x - 529595100831149643500783440460a + 462101295216340382904558146918 \)
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