ex.24.9.1.127_255_383.c
Base Field
\(F = \) 2.1.2.2a1.2 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 3\cdot 2 )\)
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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 \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{4} )x + 3b \)
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
\((\mathcal{O}_K/\mathfrak p^{ 15 })^\times/U_{\mathfrak{p}^{ 15 } }\)
:
\(\begin{array}{l}
\chi^A\left((-\mu_3 + 3)c + -\mu_3 + 3 \right) &= i^{ 0 }
\\
\chi^A\left((((a + 2)\mu_3 + (a + 2))b^{2} + (2\mu_3 + 2)b - 2\mu_3 - 2)c + ((a - 3)\mu_3 + (a - 3))b^{2} + ((3a - 1)\mu_3 + (3a - 1))b + (3a - 3)\mu_3 + 3a - 3 \right) &= i^{ 0 }
\\
\chi^A\left(3a\cdot \mu_3c + 2\mu_3b + 1 \right) &= i^{ 1 }
\\
\chi^A\left((\mu_3b^{2} + (2a + 4)\mu_3)c + 4\mu_3b + 4\mu_3 + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((2a - 2)\mu_3b^{2} - 3\mu_3b)\cdot c + (a + 2)\mu_3b^{2} + 1 \right) &= i^{ 2 }
\\
\chi^A\left((((2a - 2)\mu_3 + (3a + 2))b^{2} + ((2a + 4)\mu_3 + (a + 2))b + (2a - 3)\mu_3)c + (a + 2)\mu_3b^{2} + (2a + 4)b + 2\mu_3 + 1 \right) &= i^{ 2 }
\\
\chi^A\left((b^{2} + (2a + 4))c + 4b - 3 \right) &= i^{ 0 }
\\
\chi^A\left((((3a + 2)\mu_3 + (3a + 2))b^{2} + ((a + 2)\mu_3 + (a + 2))b - 2\mu_3 - 2)c + ((3a - 3)\mu_3 + (3a - 3))b^{2} + ((3a + 3)\mu_3 + (3a + 3))b + (3a - 3)\mu_3 + 3a - 3 \right) &= i^{ 0 }
\\
\chi^A\left((4\mu_3b^{2} + (4\mu_3 + (2a - 2))b + (2a + 2)\mu_3)c + ((2a + 2)\mu_3 + 4)b^{2} + ((2a + 1)\mu_3 + 1)b + 4\mu_3 + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((2a + 4)b + (2a + 4)\mu_3)c + (2a + 2)\mu_3b^{2} + ((2a - 3)\mu_3 + 1)b + 4\mu_3 + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((4\mu_3 + (a + 4))b^{2} + 3a\cdot \mu_3b + (4\mu_3 + 4))c + ((2a + 2)\mu_3 + 4)b^{2} + ((2a + 1)\mu_3 + 1)b + 4\mu_3 + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((2a - 2)b^{2} - 3b)\cdot c + (a + 2)b^{2} + 1 \right) &= i^{ 0 }
\\
\chi^A\left((3a\cdot \mu_3b^{2} + 4\mu_3b + (4\mu_3 + (2a - 2)))c + ((2a + 2)\mu_3 + 4)b^{2} + ((2a + 1)\mu_3 + (2a + 1))b + (2a + 4)\mu_3 + 1 \right) &= i^{ 2 }
\\
\chi^A\left(3a\cdot c + 2b + 1 \right) &= i^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (8a + 4 )x^{47} + (8a + 20 )x^{46} + 28a x^{45} + (2a + 16 )x^{44} + (28a + 16 )x^{43} + (28a + 8 )x^{42} + (8a + 4 )x^{41} + 8 x^{40} + 16 x^{39} + (20a + 4 )x^{38} + (12a + 24 )x^{37} + (28a + 24 )x^{36} + (8a + 12 )x^{35} + 20 x^{34} + (8a + 16 )x^{33} + 12 x^{32} + 20a x^{31} + (20a + 24 )x^{30} + (16a + 24 )x^{29} + (12a + 8 )x^{28} + 24a x^{27} + (20a + 8 )x^{26} + 8a x^{25} + (26a + 28 )x^{24} + (12a + 24 )x^{23} + (20a + 8 )x^{22} + (16a + 8 )x^{21} + 12 x^{20} + 24 x^{19} + 24 x^{18} + (20a + 24 )x^{17} + 16 x^{15} + (12a + 8 )x^{14} + (8a + 8 )x^{13} + (4a + 24 )x^{12} + (4a + 8 )x^{11} + (12a + 8 )x^{10} + (16a + 16 )x^{9} + (28a + 8 )x^{8} + 24 x^{7} + (24a + 24 )x^{6} + 8 x^{4} + (8a + 24 )x^{2} + (16a + 16 )x + 18a + 6 \)