ex.24.7.1.55_79_103.d
Base Field
\(F = \mathbb{Q}_2\)
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 \(x^{3} + 2 \)
Inducing Field
The inertial type \(\tau\) becomes reducible over
\(K = L(c)\), \(c\) a root of \(x^{2} - b^{3} x + (3b^{2} + (3\mu_3 + 3))b \)
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
\((\mathcal{O}_K/\mathfrak p^{ 11 })^\times/U_{\mathfrak{p}^{ 11 } }\)
:
\(\begin{array}{l}
\chi^A\left(\mu_3c + 1 \right) &= i^{ 2 }
\\
\chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 }
\\
\chi^A\left((2\mu_3 + 2)b^{2}c + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((2\mu_3 + 2)b^{2} + 2\mu_3)c + 4\mu_3 + 1 \right) &= i^{ 2 }
\\
\chi^A\left(((3\mu_3 + 3)b^{2} + 4\mu_3)c + 1 \right) &= i^{ 1 }
\\
\chi^A\left(((\mu_3 + 1)b + 2)c + (2\mu_3 + 2)b - 3 \right) &= i^{ 0 }
\\
\chi^A\left((3b + 2\mu_3)c + 2b + 4\mu_3 + 1 \right) &= i^{ 2 }
\\
\chi^A\left((\mu_3b^{2} + 4)c + 1 \right) &= i^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + 425541065674964083844296654248x^{47} + 210930288664120019483076200476x^{46} - 402589349867188316271816726632x^{45} + 600494490238535354518801748500x^{44} + 210910297614203933942462034800x^{43} - 582911971233748178620630608440x^{42} + 217436717462710749872722785928x^{41} + 544629024906731144999386387512x^{40} - 405696790881253867579187514304x^{39} - 457873151506571699669202935016x^{38} - 424976177125143706972758403088x^{37} - 224474809919091957393814948592x^{36} + 397191920430246568635000782880x^{35} + 171585033398823138920363818384x^{34} - 162177688864093420321971914976x^{33} + 566934872251531175513326853912x^{32} - 521310333498013014715353192944x^{31} - 477759076520531237425937245072x^{30} + 242736797315152615759289133736x^{29} + 137304598800809947796966098176x^{28} - 286961001015287915656197735328x^{27} + 566710139389129837626710976224x^{26} + 311788860301265304037784504400x^{25} - 252998419129585731300077173148x^{24} - 344713952625567342875816358800x^{23} + 155249200034140614726984513512x^{22} - 620213811003004622069658105104x^{21} + 169463416365886427039392951448x^{20} - 29750507066740751998165768256x^{19} - 201028230834415309008654365840x^{18} - 555472202510123137790734657392x^{17} + 263528903097592910711738824656x^{16} - 481112561922809178455353455168x^{15} - 628920981351687302582808733040x^{14} - 267922356116252690422152975520x^{13} - 403530785457946196988985250256x^{12} - 116312969307977761994765419488x^{11} - 208414424006227492549978319200x^{10} - 473769452926112018434492629568x^{9} - 442111253856014794510363068208x^{8} - 307544902085542778443409047584x^{7} + 520962549896070404279883548352x^{6} + 326727945063497430904180750896x^{5} - 557123635219281206327265086752x^{4} + 597663170028139903093512010368x^{3} - 578448121466577464606185592736x^{2} + 575978260293824889636869243040x - 132556058936220414533046016460 \)