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ex.24.10.1.131_259_387.d

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
\( v(N) = 10\)
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 \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{4} )x + b \)

Underlying Character

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

Order
4
Conductor exponent
18
Values on generators of \((\mathcal{O}_K/\mathfrak p^{ 18 })^\times/U_{\mathfrak{p}^{ 18 } }\) :
\(\begin{array}{l} \chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 } \\ \chi^A\left((((3a + 2)\mu_3 + (3a + 2))b^{2} + (-2\mu_3 - 2)b - 2\mu_3 - 2)c + ((3a - 3)\mu_3 + (3a - 3))b^{2} + ((3a - 3)\mu_3 + (3a - 3))b + (a - 3)\mu_3 + a - 3 \right) &= i^{ 0 } \\ \chi^A\left((4\mu_3b^{2} + (4\mu_3 + (2a + 2))b + (2a + 2)\mu_3)c + ((2a + 2)\mu_3 + 4)b^{2} + ((2a + 3)\mu_3 + 3)b + 4\mu_3 + 1 \right) &= i^{ 0 } \\ \chi^A\left(((2a - 2)b^{2} - b)c + (3a - 2)b^{2} + 1 \right) &= i^{ 0 } \\ \chi^A\left((\mu_3b^{2} + (2a + 4)\mu_3)c + 4\mu_3b + 4\mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left(\mu_3c + 1 \right) &= i^{ 0 } \\ \chi^A\left((b^{2} + (2a + 4))c + 4b - 3 \right) &= i^{ 0 } \\ \chi^A\left(a\cdot c - 2b + 1 \right) &= i^{ 0 } \\ \chi^A\left((a\cdot \mu_3b^{2} + 4\mu_3b + (4\mu_3 + (2a - 2)))c + ((2a + 2)\mu_3 + 4)b^{2} + ((2a + 3)\mu_3 + (2a + 3))b + (2a + 4)\mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left(((4\mu_3 + 2a)\cdot b^{2} + (4\mu_3 + 4)b + 4)c + 4\mu_3b^{2} + (-3\mu_3 + (2a - 3))b + 4\mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left((((a + 2)\mu_3 + (a + 2))b^{2} + ((a - 2)\mu_3 + (a - 2))b - 2\mu_3 - 2)c + ((a - 3)\mu_3 + (a - 3))b^{2} + ((3a + 1)\mu_3 + (3a + 1))b + (a - 3)\mu_3 + a - 3 \right) &= i^{ 0 } \\ \chi^A\left(((2a - 2)\mu_3b^{2} - \mu_3b)\cdot c + (3a - 2)\mu_3b^{2} + 1 \right) &= i^{ 2 } \\ \chi^A\left(((4\mu_3 + (3a + 4))b^{2} + 3a\cdot \mu_3b + (4\mu_3 + 4))c + ((2a + 2)\mu_3 + 4)b^{2} + ((2a + 3)\mu_3 + 3)b + 4\mu_3 + 1 \right) &= i^{ 2 } \\ \chi^A\left((((2a + 2)\mu_3 + (3a + 4))b^{2} + ((3a + 4)\mu_3 + (3a + 4))b + ((3a - 2)\mu_3 + (2a + 2)))c + ((3a - 3)\mu_3 + 2a)\cdot b^{2} + ((a - 1)\mu_3 + (a - 3))b + 3a - 1 \right) &= i^{ 1 } \end{array} \)

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (-324811737210834938235914430008a + 530336601826584742188590253352 )x^{47} + -400069096261628818286298986556a - 332009459122068356738096381720 x^{46} + (-601079916102146046696232128164a + 317737302327549879988197267680 )x^{45} + (-45452386130086510909206037172a + 47493976840895027635444917044 )x^{44} + (-404074658962161345615172775192a + 290612385598153867243944344160 )x^{43} + -248767267363515567986483766752a - 249490793302055313585601228292 x^{42} + (-39721868783915819361546693432a + 429488977284930338661282348208 )x^{41} + (537585229074284509905901297212a + 497219049448412536706039486724 )x^{40} + (-150280651556294335947410477160a + 610235778156229945365613511216 )x^{39} + (267231811755979278548451048400a + 338829538538349688118126352256 )x^{38} + -217171341479723115883408987676a - 585220431780880516287192470688 x^{37} + (412277939336532101143188090584a + 397900012866763286345408342620 )x^{36} + (499048316250131730969255204000a + 50257665340202323420483664624 )x^{35} + (-241483933291378961137681276484a + 449402230041155950930015112312 )x^{34} + -369018706758687130895012206584a - 136200448670109354250286275960 x^{33} + -460945798924360764710080741276a - 226738242287140541623736545880 x^{32} + -364968205233060103881053028128a - 579288954009160414827499364512 x^{31} + -431766309382515440746240732280a - 244821805470453740197911006640 x^{30} + -363672510494570898247958267504a - 272933299121171781038714191448 x^{29} + (-257800530754238904424669094884a + 46137553733567981024502559232 )x^{28} + (-517613307130233570608661983744a + 394623699965637175351513174848 )x^{27} + (-215236166406112118496896908956a + 153040151696804803168553542528 )x^{26} + (-156738904080599564366877624584a + 183328725873987613427543750936 )x^{25} + (336164492383278487680331019606a + 599198240677724131807789158796 )x^{24} + -227431023623877544687596783032a - 147466758957500803559691674688 x^{23} + -498772599490719239705311398632a - 447316385216462195213295257192 x^{22} + -5229946937571285702924381632a - 576476629902862121729679422872 x^{21} + -351905251433399592523288784820a - 613425962342561822785335475456 x^{20} + (349063818545875800672631316864a + 41524479143801893380299463600 )x^{19} + (-163436956756943451507541888500a + 62514931687472885303289240408 )x^{18} + (-418730471953612359701963540432a + 276267223839883187345712547136 )x^{17} + (84253164557787134733995841924a + 311889525980015999058515198192 )x^{16} + (433551262379911232135391585312a + 335734451507136815467763668480 )x^{15} + (-251190805649815372435798861480a + 151161722860209352252053694112 )x^{14} + (-250763376725504182927590344800a + 404730023809355878509548178168 )x^{13} + -243722073181173564411408225016a - 451606809307269857203907702408 x^{12} + -612577283671992985378432497824a - 67781354497151292165309144480 x^{11} + -177061429369949803103260849712a - 33299397947939835531494839160 x^{10} + (-288397965515790056179074362072a + 341116473456915706766900750176 )x^{9} + -549300908841829591802065618152a - 72768802788997388354193555928 x^{8} + 133753729069948811194202747152a - 617654801088574830145804433008 x^{7} + (180850474901435984809113008032a + 45084802928648160116691302400 )x^{6} + (-599817108393440883096569739688a + 73313825755298586330023085584 )x^{5} + (-425683680924437821200895648960a + 202780358251750811109488073544 )x^{4} + -170373650867834528137994069136a - 20710719464661344890484529568 x^{3} + 301322359114989836912105814560a - 86414742740609702745519040440 x^{2} + -352896488712081243601847701752a - 621466386115421391920698809920 x - 237321224408384138930455798566a - 7611468484088424538023823418 \)
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