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ex.24.10.1.31_63_95.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 \(x^{2} + 90546471444873528678335943241b^{3} x + (90546471444873528678335943241\mu_3 + 90546471444873528678335943241)b \)

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 \((\mathcal{O}_K/\mathfrak p^{ 20 })^\times/U_{\mathfrak{p}^{ 20 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 1 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} - 2a\cdot \mu_3)c + 4\mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left((2a\cdot \mu_3 + (2a + 2))b\cdot c + 4\mu_3b^{2} + (4\mu_3 + (2a + 4))b + a\cdot \mu_3 - 3a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 - 1)b + 2)c + (3a\cdot \mu_3 + 3a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 - 1)b^{2} - 2b + 4\mu_3)c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 1 } \\ \chi^A\left((b + 2\mu_3)c + a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + (a + 2))b + ((-3a - 2)\mu_3 + 4))c + (3a + 3)b + (-3a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((3a - 2)\mu_3 + 4)b^{2} + (2a - 2)b - 3a\cdot \mu_3 - a)c + (-\mu_3 + (2a + 4))b^{2} + (-\mu_3 + (2a + 3))b + (3a - 3)\mu_3 - 2a + 3 \right) &= i^{ 0 } \\ \chi^A\left((2b^{2} - 2\mu_3b - 2a\cdot \mu_3 - 2a)\cdot c + 3b^{2} + (2a + 1)\mu_3b + (2a + 1)\mu_3 + 2a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + 4)b^{2} + (2a + 4)b + 4a\cdot \mu_3)c + (2a + 2)\mu_3b^{2} + (\mu_3 + 4)b + (2a - 2)\mu_3 + 2a - 1 \right) &= i^{ 2 } \\ \chi^A\left((((a + 4)\mu_3 + (3a + 2))b^{2} + (-2\mu_3 + (3a + 2))b - 2\mu_3 + 4a)\cdot c + (2\mu_3 - 2)b^{2} + \mu_3b - 2\mu_3 + 2a + 3 \right) &= i^{ 0 } \\ \chi^A\left((\mu_3b^{2} + (2\mu_3 + 2)b + 4)c + 2a\cdot \mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + 2a)\cdot b^{2} + 2\mu_3)c + (4\mu_3 + 4)b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -97852459047642662183674462436a - 628524657377182367834157856480 x^{47} + (618958652988130592347842592220a + 92993501030020656283467516608 )x^{46} + 503723200344442199305709158504a - 187561565410099911749821424568 x^{45} + -137643543511525497571511242796a - 624289441746149590946271779520 x^{44} + (-540188356891770532446003517552a + 300102251602944001249709053228 )x^{43} + (-365076276292514639451303424628a + 234919873982323273578248857672 )x^{42} + 398196671558965573028831840640a - 570215286734492309055919936440 x^{41} + 252126187891833802088359547432a - 514286093414240515011059428824 x^{40} + (-23732757871882767398451110688a + 45678707093102703520836890256 )x^{39} + -396236656012925283224086526408a - 33564790215651878324430933676 x^{38} + 593184713138048327461995579264a - 259542522079903515403163761640 x^{37} + 584533265456290226958336899740a - 311898449158040002517732739904 x^{36} + (381231571839271347044370391904a + 156443893260010586839315908056 )x^{35} + (538841583657816399425785246788a + 202442616786917791226087467264 )x^{34} + (546919342553192336113768806216a + 4582892906092150285568554472 )x^{33} + (21915550253081128729346108828a + 21306144947559065799630422504 )x^{32} + 424571566717003185558119120836a - 361933576747712039149697182416 x^{31} + 183923429436443155092111829160a - 451321280764273550276244653708 x^{30} + 422389225190862044443358263760a - 53084967087990339150007910640 x^{29} + (15027011292801653071242388056a + 302201854964904631600871357104 )x^{28} + -264124243581843551865640853168a - 610940764289123557789768060896 x^{27} + 415359889623953220284098904240a - 569720294564880635840801021112 x^{26} + (313002557865054038708053290576a + 437263654077414725657863657056 )x^{25} + 577073881137380180015455201878a - 11963519013408238610093212672 x^{24} + -483300399859042755172957797424a - 292177803859065654277496093064 x^{23} + (588099704559621538706034752720a + 61233400530008170265762185928 )x^{22} + 251799730353075912115514238824a - 85676468373827308309890210320 x^{21} + 62304167348358227207727461272a - 209609813011495988946071076528 x^{20} + (-77052792063787280422360207628a + 503401265736794725535518057712 )x^{19} + -492614808205399229424016108672a - 102614588191667988703479256808 x^{18} + (552453142598107770104701848040a + 55073195030042662070466069616 )x^{17} + (-585167108505915674227455887792a + 478830729500493003286685368704 )x^{16} + 478335236680311941354629057168a - 523738330843366168483589667696 x^{15} + (-404900716614098380911233755788a + 510450670627083438285283058640 )x^{14} + -218276570517378850233469854160a - 6751022854989859792532052704 x^{13} + (92796588161564026990325292064a + 486344921475988181729376709484 )x^{12} + 205327746390171899115562379064a - 472745478497014441598509942560 x^{11} + (30865242424878532757370783216a + 27465059598224725690860912488 )x^{10} + (605141378157274895241619631288a + 614631250283092102975527591856 )x^{9} + -480526851209091318966778152424a - 45428361921265673825001504392 x^{8} + 41832347535217773338870369264a - 509249095730974990870121860680 x^{7} + 450372821465354982197726662500a - 327183590849849583069507665072 x^{6} + 225816166222177366918301686048a - 515982051838324949477155695552 x^{5} + (-218778877651564069799172654992a + 274177332811268522382011422432 )x^{4} + (435694338169930083577772775808a + 115709324156091843516147137248 )x^{3} + 239885055254685077256867779384a - 532692110211453691678569075200 x^{2} + (-418263632061584002172464950888a + 2592838172558040308481204992 )x - 254136491778544741572616575096a + 623800752269645807834359120546 \)
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