ex.24.13.1.3777_5459_7111.a
Base Field
\(F = \) 2.1.2.3a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 5\cdot 2 )\)
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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 \(x^{2} + 253530120045645880299340641075b^{6} x + ((-126765060022822940149670320538a + 3)b^{2} + (2\mu_3 + 2)b + (-a - 3)\mu_3)b \)
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
\((\mathcal{O}_K/\mathfrak p^{ 23 })^\times/U_{\mathfrak{p}^{ 23 } }\)
:
\(\begin{array}{l}
\chi^A\left(\mu_3c + 1 \right) &= i^{ 1 }
\\
\chi^A\left((((3a + 2)\mu_3 + (3a + 2))b^{2} + 2\mu_3b + 4)c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((2a\cdot \mu_3 + 2a)\cdot b + 2\mu_3)c + 1 \right) &= i^{ 0 }
\\
\chi^A\left((((3a + 3)\mu_3 + 3a)\cdot b^{2} + (3\mu_3 + 2a)\cdot b + ((3a + 2)\mu_3 + a))c + ((a + 2)\mu_3 + (2a + 1))b^{2} + ((2a + 2)\mu_3 + (2a + 1))b + (3a + 2)\mu_3 - 3 \right) &= i^{ 2 }
\\
\chi^A\left((((2a + 3)\mu_3 + (2a + 3))b^{2} + (3a + 1)\mu_3b + a)\cdot c + ((3a + 2)\mu_3 + (3a + 2))b^{2} + 3a\cdot \mu_3b + 1 \right) &= i^{ 0 }
\\
\chi^A\left((2a\cdot \mu_3b^{2} + ((3a + 1)\mu_3 + (3a + 3))b + (a\cdot \mu_3 - 2))c + (2a\cdot \mu_3 + 2)b^{2} + ((2a + 2)\mu_3 + 2)b + (2a + 4)\mu_3 + 2a + 1 \right) &= i^{ 3 }
\\
\chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 }
\\
\chi^A\left((3a\cdot b^{2} + (2\mu_3 + 2)b + (a + 2)\mu_3)c + 2a\cdot b^{2} + (2a + 4)\mu_3 + 1 \right) &= i^{ 1 }
\\
\chi^A\left(((a + 2)b^{2} + (2\mu_3 + 2)b + 4\mu_3)c + 2a\cdot b^{2} + 1 \right) &= i^{ 3 }
\\
\chi^A\left(((2a\cdot \mu_3 + (2a + 2))b^{2} + (2a + 4))c + 2\mu_3b^{2} + (a + 3)b + (a + 4)\mu_3 + a - 3 \right) &= i^{ 2 }
\\
\chi^A\left((3a\cdot \mu_3b + 4)c + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((2a\cdot \mu_3 + (2a + 1))b^{2} + (2a\cdot \mu_3 + (2a + 2))b + ((2a - 2)\mu_3 + 2))c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + 2a\cdot \mu_3b + 2a\cdot \mu_3 + 1 \right) &= i^{ 2 }
\\
\chi^A\left(((a\cdot \mu_3 + a)b + 4\mu_3)c + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((a\cdot \mu_3 + a)b^{2} + 2\mu_3b + (a + 2))c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + 2a - 3 \right) &= i^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (8a + 16 )x^{47} + (8a + 28 )x^{46} + (24a + 24 )x^{45} + (22a + 4 )x^{44} + 20a x^{43} + 12 x^{42} + (28a + 30 )x^{40} + (4a + 8 )x^{39} + 4 x^{38} + (8a + 8 )x^{37} + (30a + 20 )x^{36} + (20a + 24 )x^{35} + 16a x^{34} + (16a + 16 )x^{33} + (22a + 10 )x^{32} + (4a + 16 )x^{31} + 12 x^{30} + 16 x^{29} + (4a + 4 )x^{28} + 16 x^{27} + (12a + 20 )x^{26} + 8a x^{25} + (10a + 2 )x^{24} + (16a + 16 )x^{23} + (28a + 24 )x^{22} + (8a + 16 )x^{21} + (4a + 12 )x^{20} + (24a + 24 )x^{19} + (24a + 24 )x^{18} + (14a + 12 )x^{16} + (8a + 8 )x^{15} + (4a + 24 )x^{14} + 8a x^{13} + (16a + 4 )x^{12} + (8a + 24 )x^{11} + 16 x^{10} + 16 x^{9} + (26a + 4 )x^{8} + (24a + 8 )x^{7} + (20a + 24 )x^{6} + (20a + 24 )x^{4} + (24a + 16 )x^{3} + (12a + 16 )x^{2} + 16 x + 10a + 10 \)