ex.24.3.1.1_3_5.a
Base Field
\(F = \mathbb{Q}_2\)
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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
\( v(N) = 3\)
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 \(x^{2} - b x + b \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
4
Conductor exponent
3
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 3 })^\times/U_{\mathfrak{p}^{ 3 } }\)
:
\(\begin{array}{l}
\chi^A\left(\mu_3c + 1 \right) &= i^{ 1 }
\\
\chi^A\left(((4\mu_3 - 1)b^{2} + 4\mu_3b)\cdot c + (2\mu_3 - 2)b^{2} + (4\mu_3 + 4)b + 4\mu_3 - 1 \right) &= i^{ 0 }
\\
\chi^A\left((-3b^{2} + (-2\mu_3 + 2)b + (2\mu_3 + 2))c + (3\mu_3 + 3)b^{2} + (-3\mu_3 + 3)b + 4\mu_3 - 3 \right) &= i^{ 2 }
\\
\chi^A\left(((-2\mu_3 + 2)b^{2} + 2b + 2)c + (-2\mu_3 + 1)b^{2} + (-\mu_3 - 1)b + 3\mu_3 + 3 \right) &= i^{ 0 }
\\
\chi^A\left(((-2\mu_3 + 4)b^{2} - 2\mu_3b + 4\mu_3 + 4)c + 3b^{2} + \mu_3b + 3\mu_3 + 4 \right) &= i^{ 0 }
\\
\chi^A\left((-\mu_3 + 3)c + 4b - \mu_3 + 3 \right) &= i^{ 0 }
\\
\chi^A\left((\mu_3b^{2} - 2\mu_3b - 2\mu_3)c + 2\mu_3b + 4\mu_3 + 1 \right) &= i^{ 0 }
\\
\chi^A\left((2b^{2} - 2b + 4\mu_3 + 4)c + (-2\mu_3 - 1)b^{2} + (-\mu_3 - 3)b - \mu_3 - 1 \right) &= i^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + 24x^{47} + 300x^{46} + 2576x^{45} + 16974x^{44} + 91080x^{43} + 412896x^{42} + 1621224x^{41} + 5612805x^{40} + 17363896x^{39} + 48497064x^{38} + 123286440x^{37} + 287134346x^{36} + 615939264x^{35} + 1222297740x^{34} + 2252056776x^{33} + 3864164634x^{32} + 6190070040x^{31} + 9276875476x^{30} + 13029127584x^{29} + 17172595110x^{28} + 21263575256x^{27} + 24755608584x^{26} + 27114249962x^{25} + 27948336405x^{24} + 27114250116x^{23} + 24755609288x^{22} + 21263577698x^{21} + 17172601974x^{20} + 13029143732x^{19} + 9276907950x^{18} + 6190126686x^{17} + 3864251228x^{16} + 2252173644x^{15} + 1222437720x^{14} + 616088606x^{13} + 287276608x^{12} + 123407616x^{11} + 48589424x^{10} + 17426902x^{9} + 5651271x^{8} + 1642232x^{7} + 423150x^{6} + 95538x^{5} + 18686x^{4} + 3144x^{3} + 456x^{2} + 56x + 7 \)