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ex.24.10.1.33_67_101.d

Base Field
\(F = \) 2.1.2.2a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 3\cdot 2 )\) 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 \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{3} )x + ((-105637550019019116791391933781a - 211275100038038233582783867562)b + (-140850066692025489055189245042a + 140850066692025489055189245041))b \)

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 + 2)\mu_3 - 2)b^{2} + (4\mu_3 - 2)b - 2\mu_3 + 4a + 4)c + ((3a + 3)\mu_3 + 2a)\cdot b^{2} + ((a - 3)\mu_3 + (a - 3))b + (2a + 4)\mu_3 - a + 1 \right) &= i^{ 2 } \\ \chi^A\left((((a + 4)\mu_3 + (2a + 2))b^{2} + ((2a + 4)\mu_3 + 4)b - 2a\cdot \mu_3)c + (2\mu_3 + 2a)\cdot b^{2} + (2\mu_3 + 2)b + 3a\cdot \mu_3 - a - 3 \right) &= i^{ 2 } \\ \chi^A\left(((3a\cdot \mu_3 + (a + 3))b^{2} + ((2a - 2)\mu_3 + (3a - 1))b + (a\cdot \mu_3 + 2a))\cdot c + (\mu_3 + 3)b^{2} + ((3a + 1)\mu_3 + (2a - 1))b + (-a + 1)\mu_3 - a - 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 1)b^{2} + 2b)\cdot c + 2a\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 - 3)b^{2} + ((2a + 4)\mu_3 + (a + 4))b + ((4a + 4)\mu_3 - 1))c + ((2a + 4)\mu_3 + (2a - 3))b^{2} + (4\mu_3 + 3a)\cdot b + -2a - 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 2)\mu_3b^{2} + ((3a + 4)\mu_3 + (2a + 4))b + ((3a + 4)\mu_3 + 4a))\cdot c + ((a + 2)\mu_3 + 4)b^{2} + ((2a + 3)\mu_3 + 4)b + (-2a - 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((3a\cdot \mu_3b^{2} + (a + 4)\mu_3b + 2a\cdot \mu_3)c + (a - 2)\mu_3b^{2} + (2a + 1)\mu_3b + (-2a - 1)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + a)b^{2} + (4\mu_3 + (3a + 4))b + ((-a + 2)\mu_3 - 2a + 4))c + ((a + 2)\mu_3 - 2)b^{2} + ((2a - 1)\mu_3 + 4)b + \mu_3 + 4 \right) &= i^{ 0 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} + (2a + 4)b + (4\mu_3 + (4a + 4)))c + ((a + 2)\mu_3 + (a + 4))b^{2} + (\mu_3 + (2a - 3))b + (-2a - 2)\mu_3 - 2a - 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a - 2)\mu_3 + (2a + 2))b^{2} + (3a\cdot \mu_3 + (a + 4))b + (3a\cdot \mu_3 - a + 4))c + (-\mu_3 + 3)b^{2} + ((a - 3)\mu_3 + (a + 1))b + (-a - 3)\mu_3 + 3a - 3 \right) &= i^{ 0 } \\ \chi^A\left(((a + 2)b^{2} + (2a - 1)b + (2a - 2))c + (2a - 2)b^{2} + (3a + 2)b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((a + 2)\mu_3 - 2)b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + (2\mu_3 + (3a - 2)))c + ((2a - 1)\mu_3 + (a - 2))b^{2} + ((3a - 1)\mu_3 + (a + 3))b + (-2a + 2)\mu_3 - a + 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)\mu_3b^{2} + 2\mu_3b)\cdot c + 2a\cdot \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} + (156875814450899096738459668296a + 455673895382262948470916784544 )x^{47} + 480211144669106467745533654548a - 164946760358300741567990728396 x^{46} + 604466247332998619831968243340a - 530263505426051359688262941720 x^{45} + (-631435125834341350191178257384a + 576738033094958223913298061108 )x^{44} + (192758293701856755503514299196a + 616297425705057676791621941540 )x^{43} + 27338886280082580292811130520a - 338669823451957607973647823608 x^{42} + 3209567257336878538638774248a - 296081927028879451396329263560 x^{41} + -227084296208413382094094961348a - 209517915241600805883232288688 x^{40} + (620004013092078667115835138536a + 279937354251977382416197318256 )x^{39} + (-70570937071307954304835204716a + 399523065503871489558080359156 )x^{38} + 200721495819065180026202502604a - 414024262665437234983935166484 x^{37} + (604405169647343932761806301940a + 442074281128316439643583478416 )x^{36} + (337622579811845230669682851448a + 615341292363019497425788797032 )x^{35} + (-257825086651998314213783373980a + 92158466045593327757370646584 )x^{34} + (-556724742165003844233843054284a + 371551125962319578458820662200 )x^{33} + (-429015706543501382604313298252a + 4048502359073227183348463072 )x^{32} + (-213465686436047454769597465716a + 36993291869988445765016549584 )x^{31} + 370357146568394175995102326208a - 576452975034569904832634768732 x^{30} + (-442784659307023403586185034356a + 404840538201194055504059793184 )x^{29} + 411029678031351000554810111208a - 566478904628538253319609499624 x^{28} + (350378976981203318568279225544a + 596615839959124719212529713736 )x^{27} + (-8541570075987175889050426304a + 549786190527863325390000414124 )x^{26} + (-143865943234543556350843777372a + 626677343796250782890478404200 )x^{25} + (-382600761513941128045333596018a + 196816045937823655459794900564 )x^{24} + -195687284450457282043747352208a - 491423600912785343565653631456 x^{23} + (-144605830889848988235677509676a + 449312788696278618858701122176 )x^{22} + 30722874171019486926867048616a - 262520271503810051805595593208 x^{21} + (-195127218615068696372901349932a + 609333520599182225552722334576 )x^{20} + 363174599946925298291997647476a - 88244167331639298896938618088 x^{19} + (-366506964795628543693796128816a + 96841869797667661973458694640 )x^{18} + (-77331721455814018615461897136a + 75565404923497377918764095920 )x^{17} + -179301474317784821608667442600a - 452897210555700121887588639480 x^{16} + 115851546197919117916476259064a - 282944004248051190240564660720 x^{15} + (118743137662097856927344855388a + 6955530858111926366668454992 )x^{14} + 599440099782405985784467038668a - 591620650317198701338322987040 x^{13} + (89886027217808652452903517144a + 371646940793869737759965698028 )x^{12} + (51479486864230400501952931104a + 129383492257104728216680879872 )x^{11} + (-424110732365657939752017753560a + 22635533917113886803204668120 )x^{10} + (138823697345523764789273832904a + 55661998792249925769744142104 )x^{9} + (-371874102223260548217143556544a + 354394818479971131613554557168 )x^{8} + -366054243089405728936409372520a - 291873896648310622225377620248 x^{7} + -552983034710769594422591887500a - 28502641352333049400763097768 x^{6} + (-313275762131146062123416968304a + 46176440973306202688291052824 )x^{5} + (-565415657223944731180661461360a + 519929724982855068344063213472 )x^{4} + -415181190880044223405966481192a - 384688231374606412757579532656 x^{3} + (303353554155638378133153560348a + 510391499909356446946796571928 )x^{2} + (-499735079938001877757807285088a + 620818970039038757573013848984 )x + 40819536209063796239520001826a + 358288947935797363838420550190 \)
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