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ex.24.10.1.33_67_101.b

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 \(x^{2} - 211275100038038233582783867563b^{3} x + (-105637550019019116791391933781a\cdot b - 211275100038038233582783867563)b \)

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^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + 2)b^{2} + 2\mu_3b + 4)c + (2a\cdot \mu_3 + (2a - 2))b^{2} + ((2a + 1)\mu_3 + (2a + 1))b + 4\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 4)\mu_3b^{2} + (4a + 2)\mu_3)c + 4\mu_3b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a - 3)\mu_3 + (3a - 1))b^{2} + (2\mu_3 + (2a - 3))b + ((-a + 2)\mu_3 + (a + 2)))c + ((3a - 2)\mu_3 + (a + 4))b^{2} + ((2a - 1)\mu_3 + (a - 3))b + (2a + 4)\mu_3 - 2a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a + 1)b^{2} + (2a + 2)b)\cdot c + (2a + 4)b^{2} + 2a\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + (2a + 3))b^{2} + (a - 3)\mu_3b + ((2a - 2)\mu_3 - 2))c + (3\mu_3 + (a - 2))b^{2} + (-\mu_3 + (a - 1))b + (2a - 2)\mu_3 + a + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 2)\mu_3 + 2)b^{2} + (a\cdot \mu_3 + a)b + ((-2a + 4)\mu_3 + (3a + 4)))c + (3a\cdot \mu_3 + (a - 2))b^{2} + (3\mu_3 + 3)b + (-2a - 2)\mu_3 - 2a + 1 \right) &= i^{ 2 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} + ((a - 2)\mu_3 + (2a - 2))b + ((4a - 2)\mu_3 + (4a - 2)))c + ((3a + 4)\mu_3 + (3a + 4))b^{2} + (-3\mu_3 + (2a + 1))b - 2\mu_3 - 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 2)\mu_3 + (2a + 4))b^{2} + ((2a - 2)\mu_3 + (2a + 2))b + (4\mu_3 - 2a - 2))c + ((3a + 1)\mu_3 + 2)b^{2} + ((3a - 3)\mu_3 + (3a - 3))b + 4\mu_3 - 3a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-2\mu_3 + 4)b^{2} + ((3a + 2)\mu_3 + 4)b + (a - 2)\mu_3)c + a\cdot \mu_3b^{2} + (3a + 3)\mu_3b + (3a + 3)\mu_3 + 4a \right) &= i^{ 0 } \\ \chi^A\left(((4\mu_3 + 2)b^{2} + ((3a + 4)\mu_3 + (3a + 4))b + ((4a + 4)\mu_3 - 3a - 2))c + ((2a + 1)\mu_3 + (3a + 2))b^{2} + ((3a - 1)\mu_3 + (3a - 1))b - 2\mu_3 - a - 1 \right) &= i^{ 0 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + 2a)\cdot b^{2} + ((a + 4)\mu_3 + (3a - 2))b + (a - 2))c + ((2a - 1)\mu_3 + (3a + 2))b^{2} + ((a - 1)\mu_3 + (3a - 1))b + (-2a - 2)\mu_3 - 3a - 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a + 1)\mu_3b^{2} + (2a + 2)\mu_3b)\cdot c + (2a + 4)\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} + (-390846681065324389186856264580a + 221573999544615246987328313752 )x^{47} + (-549236306856584739425915699896a + 39732455917604004541557857340 )x^{46} + 407973084312682584966102159496a - 460837945009107850050814350480 x^{45} + (172954766038483424880103028712a + 176447726400255047386113913972 )x^{44} + -625045734703734302520696888260a - 18199737204457521175605571676 x^{43} + (-481501999119640171043936068148a + 458183856995272447185089111960 )x^{42} + (432393294343553058753118503108a + 212663921327980961823728355928 )x^{41} + -434075299806810551682845356424a - 480287584932713087321661018648 x^{40} + -348592922720161212050701123720a - 247482404435514862928487746536 x^{39} + (392502300679553372293556462364a + 610626090340771584873148466708 )x^{38} + (305441265351177752555819624272a + 527315734146903664556940479252 )x^{37} + -419843728378374719718823099552a - 301116775178578255514790368032 x^{36} + (120448301312600279386159295640a + 289489918478301342866859720400 )x^{35} + -203142187556893837472685087476a - 545655521431894282107755689416 x^{34} + (-282182108011870603163309910428a + 598036128538979866267178206264 )x^{33} + (142195937569929427923896010224a + 346652404006841696710807346984 )x^{32} + -319944560034791290806329500056a - 598696762075103549174383005216 x^{31} + 71903578023812656778546530460a - 20967246547264119199927993524 x^{30} + (417135582460433679147852280524a + 277093852761858334237107127016 )x^{29} + -103151899089974912773653927184a - 404688116614947766323839278856 x^{28} + 368164957979602279492926299032a - 257820747474840081073810219416 x^{27} + -268776802202987040632366296864a - 279802579364195007780012468780 x^{26} + (210330120314198188725237400264a + 614063804761065685537787517984 )x^{25} + (-505353082007404010738384921846a + 106634405084584741988060180348 )x^{24} + -82468451267664901120502459448a - 487175770493252811604079085064 x^{23} + (-239855817278610977943426468964a + 175451806184867750147133674936 )x^{22} + 408932273139629845958232661232a - 166606187349657499275977542848 x^{21} + (343109306701239286482500558604a + 517507375928532592200149333840 )x^{20} + (516649039213122717649487084148a + 544175211514918111473054193672 )x^{19} + (598233546782578580583113666768a + 310346024569297582494265219896 )x^{18} + (-627159500482335044418743796024a + 185779838963798122978145842280 )x^{17} + 281313388485641456861782919136a - 404529606808803920348731502880 x^{16} + -96849098393007166219295038832a - 245517155841195555853576453136 x^{15} + -50394441953718039140223544996a - 28607684691448984706636993504 x^{14} + (-503063400379401211565543318924a + 77246785371791902478551455176 )x^{13} + (424287567737468185418685231432a + 591508142864379451449852005628 )x^{12} + 76483601435366704650645669000a - 65842622352020409161334535552 x^{11} + 202174727728687313345898359552a - 458114048992625297007056668152 x^{10} + (14073798178272839235780958656a + 201814361728116169348049018920 )x^{9} + 89942055162881122667049400976a - 65293751505868247740060710104 x^{8} + 428498675999003890907106123632a - 296400305040876838491412548896 x^{7} + (-62311015650364700011875355108a + 453778639232217350927385602496 )x^{6} + (-483362128240497443298896178624a + 484658953596727490107951068856 )x^{5} + -619920065505116191879496567328a - 612207689482153573727331887728 x^{4} + -397563576703043249063569829560a - 328367590152842792984527851872 x^{3} + -345170641660463299861681738084a - 158730472787338351995561933720 x^{2} + 545104261326696469951209099304a - 258174210131967907504059353248 x - 343341119855980643183284209556a - 163991139729829281855873354370 \)
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