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ex.24.7.1.31_63_95.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
\( v(N) = 7\)
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 \(x^{2} - 211275100038038233582783867563b^{3} x + (-211275100038038233582783867563\mu_3 - 211275100038038233582783867563)b \)

Underlying Character

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

Order
4
Conductor exponent
11
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 11 })^\times/U_{\mathfrak{p}^{ 11 } }\) :
\(\begin{array}{l} \chi^A\left(\mu_3c + 1 \right) &= i^{ 1 } \\ \chi^A\left(((4\mu_3 + 4)b^{2} - 2a\cdot \mu_3)c + 4\mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left((2a\cdot \mu_3 + (2a + 2))b\cdot c + 4\mu_3b^{2} + (4\mu_3 + (2a + 4))b - 3a\cdot \mu_3 + a + 1 \right) &= i^{ 0 } \\ \chi^A\left(((3\mu_3 + 3)b + 2)c + (3a\cdot \mu_3 + 3a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((-\mu_3 - 1)b^{2} - 2b + 4\mu_3)c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 3 } \\ \chi^A\left((-3b + 2\mu_3)c + a\cdot b + 1 \right) &= i^{ 2 } \\ \chi^A\left(((2a\cdot \mu_3 + (a + 2))b + ((a - 2)\mu_3 + 4))c + (3a - 1)b + (a + 3)\mu_3 \right) &= i^{ 0 } \\ \chi^A\left((((3a - 2)\mu_3 + 4)b^{2} + (2a - 2)b + (a\cdot \mu_3 + 3a))\cdot c + (-\mu_3 + (2a + 4))b^{2} + (3\mu_3 + (2a - 1))b + (-a - 3)\mu_3 - 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left((2b^{2} - 2\mu_3b - 2a\cdot \mu_3 - 2a)\cdot c + 3b^{2} + (2a - 3)\mu_3b + (2a + 1)\mu_3 + 2a + 1 \right) &= i^{ 0 } \\ \chi^A\left((((2a + 4)\mu_3 + 4)b^{2} + (2a + 4)b + 4a\cdot \mu_3)c + (2a + 2)\mu_3b^{2} + (-3\mu_3 + 4)b + (2a - 2)\mu_3 + 2a - 1 \right) &= i^{ 2 } \\ \chi^A\left((((a + 4)\mu_3 + (3a + 2))b^{2} + (-2\mu_3 + (3a + 2))b - 2\mu_3 + 4a)\cdot c + (2\mu_3 - 2)b^{2} - 3\mu_3b - 2\mu_3 + 2a + 3 \right) &= i^{ 2 } \\ \chi^A\left((\mu_3b^{2} + (2\mu_3 + 2)b + 4)c + 2a\cdot \mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a\cdot \mu_3 + 2a)\cdot b^{2} + 2\mu_3)c + (4\mu_3 + 4)b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -124589267052306977044889693952a - 360784016118129134629243161472 x^{47} + 86886992336799375968104436176a - 291028810726081763100728644464 x^{46} + -217258664351000772308270323608a - 352486628545485535052087409088 x^{45} + (548393154072999718749602977420a + 568415873870420798737127112856 )x^{44} + (-385487199186940973339985818552a + 623428771674945750324644706664 )x^{43} + -29625382241648637764828775300a - 303347346432710122977815496340 x^{42} + -95944488474935023947464330464a - 565407017322971649367285068576 x^{41} + 254672043558586102581139405440a - 169570985293425895758993879812 x^{40} + (578341471130910203832476956856a + 378685544991357200012031429824 )x^{39} + (-427049416801530520552222405624a + 465941251319953250693734160320 )x^{38} + 178085611648716424491978897048a - 63016668620231332238594404688 x^{37} + 543070111848446719010231976020a - 388783307234123736155899273564 x^{36} + 307074839323762994415772493488a - 132880146601872374146241589936 x^{35} + -398950513962951773877244649544a - 508356873850933686421543097536 x^{34} + -621927833392670344715497555832a - 244354032490232603979654187504 x^{33} + (-73796645706033619050300942900a + 555750532895617967503729494600 )x^{32} + (-339667707753481957988151065760a + 257071838625363830310556919776 )x^{31} + (-135833620334171631234021520840a + 307335648075313424251034494232 )x^{30} + 162122118382388572027035960696a - 594750834125297492800276889232 x^{29} + 523091353348960316235135430828a - 523002751269072138123485078300 x^{28} + (-536389794609544361527676381904a + 1418154212131304137183318504 )x^{27} + (608791640082102306202032079576a + 323753029455461227036689918832 )x^{26} + -158149851815362348023361344840a - 350423370888117640068196324544 x^{25} + (-552732844119849731631161798002a + 144002135748161752870785490664 )x^{24} + -469195034254379431034436292352a - 98745776919161982881278383520 x^{23} + (44877094076396400504176449640a + 571358501758623776263147207008 )x^{22} + 336679095661341858636484890048a - 532248389529128395083293263632 x^{21} + -79755968579158215275351514496a - 49219622047513384079256986040 x^{20} + -376024383248777379148832910200a - 621972541732723043377236202560 x^{19} + 624432211792504607648714442680a - 4406222593564497644141039688 x^{18} + (-453414915489655125964492869200a + 380101018014636877992664280560 )x^{17} + -557800224787290976308440622620a - 231983630825009217945998628304 x^{16} + -417610986146263577699570422240a - 358678692994219520952192629264 x^{15} + 322737055528413314872648519480a - 190666764485238937905185883600 x^{14} + -605971363892050556384201451680a - 61227160629922566027242878400 x^{13} + -612364455331098662650001424120a - 332965619832531671119887059292 x^{12} + (-503627238470115658485815308544a + 396396841881019575465797564544 )x^{11} + 79531756295722500227144933616a - 534066729298693847164248170184 x^{10} + (-199113156339928291441777105448a + 403496936473176384344827407888 )x^{9} + (-175945951213318173665413062176a + 338234416883116257278651375976 )x^{8} + (560815842181790874764353289312a + 609871454219718766715171232000 )x^{7} + (-543427068728522554299965815272a + 100515774315268479838345376832 )x^{6} + (245804845729680077977324615152a + 432347550012692922176941221792 )x^{5} + 621261741311810494605401089284a - 186078512520389827738636306032 x^{4} + (586779656138799422865846169360a + 390771944893185744196090686704 )x^{3} + (163405590369391874769853968784a + 439390515740341434836531656160 )x^{2} + -127307584811417960551327359976a - 293308731722056506280157159040 x + 9178982490605405891733932076a - 589325768470375474503861095322 \)
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