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ex.24.10.1.31_63_95.a

Base Field
\(F = \) 2.1.2.2a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 2 )\) 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 \(x^{2} + ((a + 1)b^{3} )x + (-\mu_3 - 1)b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-239840388203287867108183634236a + 382068097111772142041242661776 )x^{47} + (-164224861273586979941631710140a + 299239976502202959846302357288 )x^{46} + (227020404281829122055123085216a + 411055632963190751292523990272 )x^{45} + -599770439392772828604325631088a - 550731185555526611508396048432 x^{44} + -198495707702466647621876362120a - 472123971315389818838172327252 x^{43} + -562744708098705705786627882604a - 144109114590014412024790848968 x^{42} + (-5076353133443074472749980180a + 504555705537971337248701978320 )x^{41} + (-275675146595340285240688571504a + 32540137676460967316574547824 )x^{40} + (294733970605706075235759047160a + 513817838647290895452222519200 )x^{39} + (527633027290085943117626454704a + 363025292763066516972946562284 )x^{38} + (-431772713186644507345388147304a + 84771955175776699514388487400 )x^{37} + (402141071975063148645352130772a + 71139361529608095449175872144 )x^{36} + (80074487117943622480797225136a + 441637368641364659355862637552 )x^{35} + (584365160965058778464523295656a + 67013896080719632001162292016 )x^{34} + (-214990645003833800300742975512a + 267668945741671508315873970056 )x^{33} + 67776293922571919799358694080a - 283020266248817942700820702632 x^{32} + -630723547731161350212175090068a - 293994496286500609410831490576 x^{31} + (58530480118731613919623658932a + 425381002021009862941911194700 )x^{30} + (209830268815571795214284997284a + 47358661554381312670207360496 )x^{29} + (389269486607862429143025244424a + 315639470031913269463064494440 )x^{28} + 337938641431228145644040809600a - 288767680583083193223705840608 x^{27} + (85053714651108061910856167976a + 261636718940821772490576680728 )x^{26} + (415307982848833439266962193608a + 303521303002996053860978042664 )x^{25} + -95263649333721983183294545218a - 320512178934304775296917248952 x^{24} + -141978497309181293557997562016a - 92064966277396295722823660440 x^{23} + (181225847704072349948895012384a + 476006823405197617549502197064 )x^{22} + (-477696990121518132037956181632a + 43822359554500508581190184000 )x^{21} + 239938239569318109530824144560a - 323194946352958866027910783352 x^{20} + (545060927207734343766706631988a + 235988042984727759025166105232 )x^{19} + (421148678587075038725247660344a + 23453547540593893054761787784 )x^{18} + 607287033592582964055476347896a - 553352644180093532760191029368 x^{17} + (-280653277454151286029699455088a + 587994713381915383460773626768 )x^{16} + 600084805625490912936061088816a - 421252980992210015963068438304 x^{15} + 359549293116130622955804506612a - 256754753734027444506870535152 x^{14} + -216208028037644465942775771136a - 377531115485113295072621318080 x^{13} + (-14275661486625307104788403336a + 444733745446563157629442833060 )x^{12} + (-378797085246276914701912848248a + 494242545288350996730924918800 )x^{11} + (71625816241886517651285360544a + 82315798478723668938216679376 )x^{10} + -84343864955565483112139032408a - 522920435250308650716441216400 x^{9} + (291291012872031125013870445368a + 517419432881500072167179677712 )x^{8} + 390251545242911520837299140360a - 592422235765860067042702832472 x^{7} + (631798073355085666261184340580a + 88778618530503756134615337192 )x^{6} + -468732611269147921907868933784a - 261001067248621881253502372056 x^{5} + (149067140746361864474648723320a + 487243189369192871829537165408 )x^{4} + (456631327360889353799436253024a + 111958415355699194007342559232 )x^{3} + -533841456095291117035274814520a - 148140334779349935493586940896 x^{2} + -578554126686343429461065654320a - 190675374991644856040897189312 x + 379721622526150406771038302110a - 25180872649629289920330796162 \)
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