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ex.24.10.1.127_255_383.d

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 \(x^{2} - 211275100038038233582783867563b^{4} x + b \)

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

Inertia Polynomial

The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + -592169270313875550061727782928a - 127521892742254812429171944152 x^{47} + (141446612777224048580161169492a + 564353116658812439515895115580 )x^{46} + (-371866803990390523539752922548a + 240394156569745772523242887488 )x^{45} + (-508229407919846647056231313972a + 179408429906319309018489776392 )x^{44} + -565102500143610919151563350048a - 513271982695318505732058011024 x^{43} + (5906737367891873080570652772a + 481097601416743917693233972364 )x^{42} + -357605418775519569622325076624a - 633620322499848920822832211528 x^{41} + 457772886992588931947565092044a - 434801140947895805931364283004 x^{40} + (-545076748463473319577712093144a + 386660697743267890739664321040 )x^{39} + 342381431293022265014547761192a - 578513089159858381267596569328 x^{38} + -569157930892188593964607187804a - 310791711299819495384863901816 x^{37} + (613297153130572352034888821012a + 548795330623453518730352645532 )x^{36} + (-527161726048593645241386828760a + 538593134095168358619916952672 )x^{35} + (552142677489379153511792994032a + 557831012529412148666770121424 )x^{34} + (-331989637722475588202086887928a + 279183621649359344455759283080 )x^{33} + (-13068834917900222736017472420a + 519684608094357264335969134128 )x^{32} + (-291502973981738766005679233600a + 211898784972910099998602538720 )x^{31} + (-93783442758958963654085701792a + 82262714686871845233455009520 )x^{30} + 237907803793567040949067433080a - 494513839641198599182419615792 x^{29} + (-229916983764524856860758935344a + 31902754733744788397144954016 )x^{28} + (79422259614917579976755007296a + 309742693381373096155246574592 )x^{27} + -131581670356036368865774142600a - 585095689743313936267220503720 x^{26} + (-18841087275422263031514433344a + 534721897317067386896958395496 )x^{25} + (166448546888426225545709680942a + 632321100591143765994037021968 )x^{24} + -437805600013016551049779369416a - 82765384720626763259030513024 x^{23} + -269450408663155594254019875004a - 633057266035826726264422869720 x^{22} + (181642856757973653948514447200a + 40373129950438346637776772776 )x^{21} + (-85321415783221913404551260632a + 178749158962783101443106370872 )x^{20} + (-436977397426555057680984525152a + 114578375953077857736983828272 )x^{19} + (-302282088433607871342106433236a + 60807893330847348338668965528 )x^{18} + -312746966150882888306440439656a - 530766873365449516812653134880 x^{17} + 629759923934366935098123203788a - 6489082327791860050238882456 x^{16} + (-254069993273305230678654460752a + 175053023039759916928806543648 )x^{15} + (-537998300481725072112795071328a + 462325806195557887100828715888 )x^{14} + -199832693262854882941206634712a - 582720784076615949363052188008 x^{13} + -169804317582629636656640895976a - 535768129658290028198550385232 x^{12} + 563776232023895492336483291216a - 221704749569140576909798871152 x^{11} + (384927425909845736951509319576a + 385820768873883868404720436992 )x^{10} + 468478859563103201026092121048a - 138793615728289964547477391344 x^{9} + -145728647454752276419452727296a - 32249682763662348316553402152 x^{8} + 40791841680708280653103627392a - 362393084272477286533160571152 x^{7} + 430024328375672621073240324048a - 216002117098362481669251681440 x^{6} + 281373466809084636782775171456a - 275338597647174522401559412336 x^{5} + 528954336330650045168871154032a - 411802577557387230854220963712 x^{4} + 473793592405998393828545178384a - 84421910255560131471605378016 x^{3} + -524788700384068658417117115704a - 313126581549411442312406734112 x^{2} + 218639662507056440105679926264a - 607702642066601925010855763008 x - 525378756922871160739213182936a - 574932743987175733824360276750 \)
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