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ex.24.8.1.31_63_95.b

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
\( v(N) = 8\)
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 \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{3} )x + ((-140850066692025489055189245042a + 140850066692025489055189245041)\mu_3 - 140850066692025489055189245042a + 140850066692025489055189245041)b \)

Underlying Character

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

Order
4
Conductor exponent
14
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 14 })^\times/U_{\mathfrak{p}^{ 14 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 3 } \\ \chi^A\left(((2a + 2)b^{2} + (-2\mu_3 - 2)b + ((4a - 2)\mu_3 + 4))c + ((3a + 1)\mu_3 + (2a - 1))b^{2} + ((3a + 3)\mu_3 + (2a + 2))b + 2a\cdot \mu_3 + 3a - 3 \right) &= i^{ 2 } \\ \chi^A\left((((3a - 2)\mu_3 + (a + 4))b^{2} + (4\mu_3 + 2a)\cdot b + 4a\cdot \mu_3)c + ((2a + 2)\mu_3 + (2a - 2))b^{2} + 4b + (a + 4)\mu_3 + a - 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)b + (2a - 2)\mu_3)c + (a - 2)b + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a - 1)\mu_3 + (2a - 1))b + (2a - 2))c + ((3a + 2)\mu_3 + (3a + 2))b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a - 3)\mu_3 + (a + 1))b^{2} - 3b + (-a + 3)\mu_3 - a)c + ((a + 3)\mu_3 + 3)b^{2} + ((a + 4)\mu_3 + (2a + 1))b + (3a - 3)\mu_3 + 4a \right) &= i^{ 2 } \\ \chi^A\left((4\mu_3b^{2} + ((2a + 4)\mu_3 + (3a + 4))b - a\cdot \mu_3 + 4a)\cdot c + 4\mu_3b^{2} + (4\mu_3 + (2a - 3))b + (-2a - 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((3a - 2)\mu_3 + 4)b^{2} + (a - 2)b + ((4a + 2)\mu_3 + (2a - 2)))c + ((a + 1)\mu_3 + (2a - 1))b^{2} + ((3a + 3)\mu_3 + 2)b + 4a\cdot \mu_3 + a - 3 \right) &= i^{ 2 } \\ \chi^A\left((((a + 4)\mu_3 + (a + 4))b^{2} + (2a + 4)b + (-2a - 2)\mu_3)c + (-2\mu_3 - 2)b^{2} + (2a - 1)b + (4a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + 2a)\cdot b^{2} + (2a + 4)b + (4\mu_3 + (4a + 4)))c + ((2a - 2)\mu_3 + 4)b^{2} + ((2a + 1)\mu_3 + 2a)\cdot b + (2a - 2)\mu_3 - 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left(((4\mu_3 + (2a + 4))b^{2} + ((a + 2)\mu_3 + (3a + 4))b + ((-3a + 4)\mu_3 + 2a))\cdot c + (2a\cdot \mu_3 + 4)b^{2} + (2a + 1)b + \mu_3 + 4 \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 - 1)b^{2} + (2a + 2)b + 4\mu_3)c + ((2a + 4)\mu_3 + (2a + 4))b^{2} + 2a\cdot b + 1 \right) &= i^{ 1 } \\ \chi^A\left((((3a + 4)\mu_3 + 3a)\cdot b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + (2a\cdot \mu_3 + 4a))\cdot c + (4\mu_3 + 4)b^{2} + (4\mu_3 + (2a + 3))b + (-2a - 1)\mu_3 + 4a \right) &= i^{ 0 } \\ \chi^A\left((\mu_3b^{2} + ((2a - 2)\mu_3 + (2a - 2))b + 4)c + (2a + 4)\mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 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} + 490438387088150914502990015284a - 181060457920262755630519004600 x^{47} + -414605236556044972594925910160a - 510558723896738628358215564960 x^{46} + (-212848267003620046444200365932a + 591285266123055091163254161288 )x^{45} + (-209796694803381286826211835960a + 113377636782904870856154701324 )x^{44} + (74329295037048839084468280396a + 574018508193359656594979868792 )x^{43} + -20671580879838982474366363104a - 407497309099459659009430255188 x^{42} + 100209327005816173566836395620a - 212240438872480098847835528568 x^{41} + (226764598590512220017675456420a + 99854009787041117177790770412 )x^{40} + -27019317562113016143286597072a - 237953611976331905920066432784 x^{39} + (467979879662055730849989544468a + 519623218843922398180128293556 )x^{38} + (-9479530720272456057576520452a + 197803427992640695859786737836 )x^{37} + (211075918730602441646856731944a + 97423148571464612373636234884 )x^{36} + 451814508663767404096371359976a - 367847714178411872938127539216 x^{35} + (376186251081520071726460064580a + 594611878407104222732496257560 )x^{34} + (-627838008919818747777738565384a + 314865169687587293756056044400 )x^{33} + (586996039996151045337607067512a + 410686556981006871526327530456 )x^{32} + -55091537856782119388449111736a - 110163987491097001224986683660 x^{31} + -544976592527974683512193547988a - 199109647496598116525741178276 x^{30} + 277637767895913741906056056536a - 475317832911954239454485242264 x^{29} + -473754771202659344225481192892a - 140229447046834081811802007232 x^{28} + 599127410487381730934774580928a - 513292510219262039716720017200 x^{27} + -570134880866371998118952138704a - 433722446539691243931999951824 x^{26} + -265528938330331884004881392660a - 411663626365320814040153518296 x^{25} + (-491257443967217694997706643482a + 464536544571081808678902023144 )x^{24} + (-106514279611673297331480753968a + 334933886292809702361753812760 )x^{23} + 504336823859273696554152101456a - 604335283245819890305431905224 x^{22} + -152246345070238525265747785328a - 396242432944607813658027396136 x^{21} + 145147974683777430411322701188a - 196279645739556174031077348672 x^{20} + -453274895283022283930922132160a - 217992703433840076690522243912 x^{19} + (103282909253711561213284996564a + 167342188141378366917399135768 )x^{18} + 519269869791524838084879526192a - 281942833748236567107244234344 x^{17} + -120574789882256366130752290100a - 420046940198964101811065102272 x^{16} + (323423802188866662103928654624a + 288194763972040842234861032288 )x^{15} + (-607655801351275783458817094604a + 367866339273472056855678686944 )x^{14} + (-66451840494408605585225755348a + 478479156457408781853415901736 )x^{13} + (234495966733093418841811386068a + 122914697558882569991524548324 )x^{12} + (-303412536890144876748214788704a + 515141452477105790644277440912 )x^{11} + 426851193782060235500382765672a - 60459090705488401812546203720 x^{10} + 262316209653628336948416936992a - 317658558589269582569266601312 x^{9} + 179897839613711546832754033328a - 494786682658167266262107318544 x^{8} + (-500918189890449663275443532708a + 130875472270789324450503794408 )x^{7} + 51076331616872116562551710348a - 364518130113846627362761386616 x^{6} + (41625684440591040386942762072a + 388264021431819687428534033584 )x^{5} + (93804124686267242264053271112a + 211323603339850480427833029272 )x^{4} + 298245138112942992233144482032a - 433663594962753468152139129520 x^{3} + -152318907516018321914496710080a - 300137789602878049393154576800 x^{2} + (-512955018317584296790369003472a + 593741506217257170866018371832 )x - 302078919783850003539697646246a - 119133930361889086004963081722 \)
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