ex.24.13.1.3647_5199_6727.e
Base Field
\(F = \) 2.1.2.2a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 2 )\)
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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
\( v(N) = 13\)
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 \(x^{2} + ((a + 1)b^{6} )x + ((-3a - 6)b^{2} + (-3a - 3)b + 1)b \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
4
Conductor exponent
23
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 23 })^\times/U_{\mathfrak{p}^{ 23 } }\)
:
\(\begin{array}{l}
\chi^A\left(\mu_3c + 1 \right) &= i^{ 2 }
\\
\chi^A\left((a\cdot b^{2} + (2a + 2))c + 2a\cdot b^{2} - 3 \right) &= i^{ 0 }
\\
\chi^A\left(((a + 2)b^{2} + (3a + 2)b + (a + 2))c + 2a\cdot b^{2} + 2a\cdot b + 2a - 3 \right) &= i^{ 0 }
\\
\chi^A\left(((3a + 3)b^{2} + 3b + 2)c + (2a + 2)b^{2} + 2b - 3 \right) &= i^{ 0 }
\\
\chi^A\left(((a + 2)\mu_3b^{2} + (3a + 2)\mu_3b + (a + 2)\mu_3)c + 2a\cdot \mu_3b^{2} + 2a\cdot \mu_3b + (2a + 4)\mu_3 + 1 \right) &= i^{ 2 }
\\
\chi^A\left(((3a + 3)\mu_3b^{2} + 3\mu_3b + 2\mu_3)c + (2a + 2)\mu_3b^{2} + 2\mu_3b + 4\mu_3 + 1 \right) &= i^{ 1 }
\\
\chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 }
\\
\chi^A\left(((2a + 2)b^{2} + 3a\cdot b + 4)c + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((2a + 2)\mu_3b^{2} + 2a\cdot \mu_3b + (2a + 2)\mu_3)c + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((2a + 2)\mu_3b^{2} + 2a\cdot b + (2a + 4))c + ((3a + 1)\mu_3 + (3a + 1))b^{2} + (\mu_3 + 1)b + 2\mu_3 - 1 \right) &= i^{ 2 }
\\
\chi^A\left(((2a + 2)\mu_3b^{2} + 3a\cdot \mu_3b + 4\mu_3)c + 1 \right) &= i^{ 3 }
\\
\chi^A\left((((3a + 2)\mu_3 + (2a + 3))b^{2} + (2a + 2)b + (3a\cdot \mu_3 + a))c + (2\mu_3 + (2a + 3))b^{2} + (\mu_3 + (2a + 3))b + 2a\cdot \mu_3 + 3a - 1 \right) &= i^{ 0 }
\\
\chi^A\left((a\cdot \mu_3b^{2} + (2a + 2)\mu_3)c + 2a\cdot \mu_3b^{2} + 4\mu_3 + 1 \right) &= i^{ 1 }
\\
\chi^A\left((\mu_3b^{2} + a\cdot \mu_3b + 2a\cdot \mu_3)c + 1 \right) &= i^{ 3 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (28a + 8 )x^{47} + 16a x^{46} + (16a + 8 )x^{45} + (20a + 28 )x^{44} + (16a + 28 )x^{42} + 4a x^{41} + (2a + 24 )x^{40} + 12a x^{39} + (4a + 16 )x^{38} + (4a + 24 )x^{37} + 18a x^{36} + (8a + 8 )x^{35} + (28a + 20 )x^{34} + (16a + 8 )x^{33} + (8a + 10 )x^{32} + 4a x^{31} + (20a + 28 )x^{30} + (8a + 8 )x^{29} + (20a + 28 )x^{28} + (24a + 24 )x^{27} + (24a + 12 )x^{26} + 24a x^{25} + (24a + 2 )x^{24} + (8a + 24 )x^{23} + (8a + 8 )x^{22} + 16a x^{21} + (24a + 24 )x^{20} + 8a x^{19} + (16a + 16 )x^{18} + (24a + 24 )x^{17} + (4a + 28 )x^{16} + (8a + 8 )x^{15} + (24a + 24 )x^{14} + (24a + 8 )x^{13} + (12a + 4 )x^{12} + (8a + 16 )x^{11} + (4a + 24 )x^{10} + (8a + 16 )x^{9} + 10a x^{8} + 24 x^{7} + (20a + 8 )x^{6} + 8a x^{5} + 20a x^{4} + 16 x^{3} + (4a + 24 )x^{2} + (16a + 16 )x + 12a + 30 \)