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ex.24.10.1.131_259_387.c

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
\( v(N) = 10\)
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 \(x^{2} - 211275100038038233582783867563b^{4} x + 3b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -463934126403983049750416699568a - 266935392037346070251220287384 x^{47} + (341713213018937471281666045224a + 27622426564238294938163801404 )x^{46} + (24724984304297662170008889620a + 541159915882097165549565311280 )x^{45} + (-499135677200885122772417552248a + 599903685086436571275238704016 )x^{44} + (-514774125117783245920197835688a + 79958893412148085695664464112 )x^{43} + -480890458420406009500859270820a - 479738045291526900594732465348 x^{42} + (-353157722824093628831073243816a + 236759225026804883839518149736 )x^{41} + (282630755559238980911758101256a + 145298821236163615922793387580 )x^{40} + -257606474180142807114747709000a - 491670480810791091406935865952 x^{39} + 417761735584210802285993135784a - 504665151624941509956191568328 x^{38} + -152734652240661892143548722548a - 129365673764632062217307103624 x^{37} + (207058205280146992926173844396a + 515849358787032975601968468188 )x^{36} + 497791319487225562761510711072a - 204540906919069637658339553472 x^{35} + -454301208352985079481335407444a - 231306794147939799270508755344 x^{34} + (335119996182472269605333170592a + 241418515635709730497684515992 )x^{33} + -236440217371518909436108011104a - 69366454974271927310559089696 x^{32} + 594665879119255360697729827840a - 482514396844400553704578940032 x^{31} + (546302874462033144747308082480a + 93532481151864043621542628160 )x^{30} + 566711100034096519590799604592a - 51667563105523206384874946704 x^{29} + (49198909076456790585210336264a + 340359330096243749217565316784 )x^{28} + (315055714218417663536111418464a + 615273068646848636850379599328 )x^{27} + 137222101813090572682462483744a - 372567730054175979924197123336 x^{26} + (609478392582848593227740600120a + 367216102708990211372677278784 )x^{25} + (584396199034052456857322694514a + 164718015705448500983055282788 )x^{24} + (-451385786017805656706303657480a + 528102835075422786675611194992 )x^{23} + -351578796485650339322607561604a - 193525289414758376037480229160 x^{22} + (-537362390057424225201744429496a + 130616192642893841745949790792 )x^{21} + (-522626633643497901755846744944a + 57782313864786992128160527456 )x^{20} + 216400814417559452981281451936a - 555922687419970727690414849456 x^{19} + -305069731793464485756794451876a - 503418045894804095191740109328 x^{18} + (-588604924298779972233547628936a + 161854943122467138543613645504 )x^{17} + 549233076860876315909296027820a - 201661709828870471951579536440 x^{16} + -188720141961673922644682283520a - 32790671073501876989608349344 x^{15} + 25769970161928131882693089208a - 199282610678569246281896885568 x^{14} + 6552716389407576814040285456a - 603290535222689821581821112312 x^{13} + (-599154029441536406077395329720a + 558360787445241748200324890384 )x^{12} + -312613811610486315156339368048a - 240846494638574649686786095360 x^{11} + (331100675222156403160796409024a + 534295980110264673696014108776 )x^{10} + 24521844934651061803679777784a - 273155663096544651800328203184 x^{9} + 617300119809355234433802628976a - 288608289123545765919124476416 x^{8} + (495115709669601278929010753824a + 215193267138557823048378697776 )x^{7} + 412887349632628281402101931728a - 511521508461216465928568433152 x^{6} + -394747212992502341340800788528a - 490876924139955337010371060192 x^{5} + -292007128621086953437123203280a - 190101228913731685301193587248 x^{4} + 346697068440896441737005582704a - 446357697085358712632197189792 x^{3} + -460234465305702213232521098232a - 177669374017002465679515810800 x^{2} + 292938930717423944695957908816a - 627630022854409617001987777840 x - 66137243412021215383775792764a - 57359613739953729771138630906 \)
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