ex.24.3.1.1_3_5.a
Base Field
\(F = \) 2.2.1.0a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + x + 1 )\)
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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:
\( L = F(b) \), \(b\) 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
\((\mathcal{O}_K/\mathfrak p^{ 3 })^\times/U_{\mathfrak{p}^{ 3 } }\)
:
\(\begin{array}{l}
\chi^A\left(a\cdot c + 1 \right) &= i^{ 1 }
\\
\chi^A\left(((4a - 1)b^{2} + 4a\cdot b)\cdot c + (2a - 2)b^{2} + (4a + 4)b + 4a - 1 \right) &= i^{ 0 }
\\
\chi^A\left((-3b^{2} + (-2a + 2)b + (2a + 2))c + (3a + 3)b^{2} + (-3a + 3)b + 4a - 3 \right) &= i^{ 2 }
\\
\chi^A\left(((-2a + 2)b^{2} + 2b + 2)c + (-2a + 1)b^{2} + (-a - 1)b + 3a + 3 \right) &= i^{ 0 }
\\
\chi^A\left(((-2a + 4)b^{2} - 2a\cdot b + 4a + 4)c + 3b^{2} + a\cdot b + 3a + 4 \right) &= i^{ 0 }
\\
\chi^A\left((-a + 3)c + 4b - a + 3 \right) &= i^{ 0 }
\\
\chi^A\left((a\cdot b^{2} - 2a\cdot b - 2a)\cdot c + 2a\cdot b + 4a + 1 \right) &= i^{ 0 }
\\
\chi^A\left((2b^{2} - 2b + 4a + 4)c + (-2a - 1)b^{2} + (-a - 3)b - a - 1 \right) &= i^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{24} + 2a x^{12} + 2 x^{9} + 2 x^{6} + 2 \)