ex.24.9.1.131_259_387.d
Base Field
\(F = \) 2.1.2.3a1.4 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 3\cdot 2 )\)
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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
\( v(N) = 9\)
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 \(x^{2} - 211275100038038233582783867563b^{4} x + 3b \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
4
Conductor exponent
15
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 15 })^\times/U_{\mathfrak{p}^{ 15 } }\)
:
\(\begin{array}{l}
\chi^A\left((2\mu_3b^{2} - 3\mu_3b)\cdot c + a\cdot \mu_3b^{2} + 1 \right) &= i^{ 2 }
\\
\chi^A\left(3a\cdot \mu_3c + 2\mu_3b + 1 \right) &= i^{ 3 }
\\
\chi^A\left(2\mu_3b^{2}c + 1 \right) &= i^{ 0 }
\\
\chi^A\left((((3a + 4)\mu_3 + 4)b^{2} + ((a - 3)\mu_3 + (a + 3))b - \mu_3 + 3a)\cdot c + (2\mu_3 + (a - 1))b^{2} + ((2a + 3)\mu_3 + (3a - 1))b + (2a + 2)\mu_3 + 2a + 1 \right) &= i^{ 1 }
\\
\chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 }
\\
\chi^A\left((((2a + 2)\mu_3 + 2a)\cdot b^{2} + ((2a + 3)\mu_3 + (3a - 2))b + ((2a + 3)\mu_3 + (a - 2)))c + ((2a - 1)\mu_3 + 3)b^{2} + ((2a + 4)\mu_3 + (2a + 4))b + (a - 2)\mu_3 + 3a - 3 \right) &= i^{ 1 }
\\
\chi^A\left((2a\cdot b + 2a\cdot \mu_3)c + (2a\cdot \mu_3 + (2a + 4))b^{2} + (3\mu_3 + (2a + 3))b + 1 \right) &= i^{ 0 }
\\
\chi^A\left((3a\cdot \mu_3b^{2} + 4b + (4\mu_3 + 2))c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + (-\mu_3 + 3)b + 2a\cdot \mu_3 + 1 \right) &= i^{ 2 }
\\
\chi^A\left(((4\mu_3 + 4)b^{2} + (-2\mu_3 + 2)b + 2)c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + (-\mu_3 + (2a + 3))b + 1 \right) &= i^{ 0 }
\\
\chi^A\left((b^{2} + 2a)\cdot c + 4b - 3 \right) &= i^{ 0 }
\\
\chi^A\left(3a\cdot c + 2b + 1 \right) &= i^{ 0 }
\\
\chi^A\left((2b^{2} - 3b)\cdot c + a\cdot b^{2} + 1 \right) &= i^{ 0 }
\\
\chi^A\left((((a - 2)\mu_3 + (a - 2))b^{2} + ((3a - 2)\mu_3 + (3a - 2))b + ((2a + 2)\mu_3 + (2a + 2)))c + ((3a - 3)\mu_3 + (3a - 3))b^{2} + ((a - 1)\mu_3 + (a - 1))b + (a - 3)\mu_3 + a - 3 \right) &= i^{ 0 }
\\
\chi^A\left(((3a + 4)\mu_3b^{2} + (-2\mu_3 + 4)b + (2a\cdot \mu_3 + (a - 2)))c + ((2a + 4)\mu_3 + 2a)\cdot b^{2} + (4\mu_3 + (2a + 2))b + 4\mu_3 + 1 \right) &= i^{ 2 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + 28a x^{47} + (16a + 16 )x^{46} + (22a + 28 )x^{44} + 16 x^{43} + 24a x^{42} + (16a + 24 )x^{41} + 20 x^{40} + 24a x^{39} + (20a + 8 )x^{38} + (24a + 16 )x^{37} + (20a + 16 )x^{36} + (12a + 28 )x^{35} + (8a + 24 )x^{34} + (8a + 16 )x^{33} + (24a + 12 )x^{32} + 28a x^{31} + (8a + 16 )x^{30} + (28a + 16 )x^{29} + (28a + 24 )x^{28} + 24a x^{27} + (8a + 16 )x^{26} + (14a + 16 )x^{24} + (8a + 24 )x^{23} + 16a x^{22} + 16a x^{21} + (4a + 20 )x^{20} + 16a x^{19} + (8a + 16 )x^{18} + 16 x^{17} + (4a + 16 )x^{16} + 16 x^{15} + 16 x^{14} + (4a + 16 )x^{12} + (28a + 24 )x^{11} + (16a + 16 )x^{10} + 12a x^{8} + (16a + 24 )x^{7} + 16a x^{6} + (24a + 24 )x^{5} + (24a + 24 )x^{4} + 16a x^{3} + (16a + 16 )x^{2} + 16 x + 8a + 10 \)