ex.24.7.1.31_63_95.a
Base Field
\(F = \) 2.1.2.3a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 )\)
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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
\( v(N) = 7\)
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 \(x^{2} - b^{3} x + (-\mu_3 - 1)b \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
4
Conductor exponent
11
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 11 })^\times/U_{\mathfrak{p}^{ 11 } }\)
:
\(\begin{array}{l}
\chi^A\left(\mu_3c + 1 \right) &= i^{ 1 }
\\
\chi^A\left(((4\mu_3 + 4)b^{2} - 2a\cdot \mu_3)c + 4\mu_3 + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((2a + 4)\mu_3 + (2a + 2))b\cdot c + 4\mu_3b^{2} + (4\mu_3 + (2a + 4))b + (a + 2)\mu_3 + 3a - 3 \right) &= i^{ 0 }
\\
\chi^A\left((-b - 2\mu_3)c + a\cdot b + 1 \right) &= i^{ 0 }
\\
\chi^A\left((\mu_3b^{2} + (2\mu_3 + 2)b + 4)c + 2a\cdot \mu_3b^{2} + (2a\cdot \mu_3 + 2a)\cdot b + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((-2\mu_3 + (2a - 2))b^{2} + (-3\mu_3 + (2a + 3))b + ((a + 4)\mu_3 - 2a - 1))c + ((a - 2)\mu_3 + (a + 2))b^{2} + ((2a + 3)\mu_3 + (3a + 1))b + (-2a - 2)\mu_3 - 3a - 3 \right) &= i^{ 2 }
\\
\chi^A\left(((2a + 4)\mu_3b^{2} + 2a\cdot b + (4a\cdot \mu_3 + 4a))\cdot c + ((2a + 2)\mu_3 + 4)b^{2} + (3\mu_3 + 4)b + (-2a + 2)\mu_3 + 2a + 3 \right) &= i^{ 2 }
\\
\chi^A\left((((3a + 4)\mu_3 + (3a + 2))b^{2} + (3a\cdot \mu_3 + (a - 2))b + ((2a - 2)\mu_3 + (3a + 2)))c + 4b^{2} + (2a + 1)\mu_3b + (-2a - 2)\mu_3 + 2a - 3 \right) &= i^{ 2 }
\\
\chi^A\left((-2b^{2} - 2\mu_3b - 2a\cdot \mu_3 - 2a)\cdot c + 3b^{2} + (2a - 1)\mu_3b + (-2a - 3)\mu_3 - 2a - 3 \right) &= i^{ 0 }
\\
\chi^A\left((((2a + 4)\mu_3 + 2a)\cdot b^{2} + (3a + 2)b + (a + 2)\mu_3)c + (3a + 1)b + (3a + 3)\mu_3 \right) &= i^{ 0 }
\\
\chi^A\left((((3a + 4)\mu_3 + (a + 2))b^{2} + (2\mu_3 + (3a - 2))b + 2\mu_3)c + (2\mu_3 - 2)b^{2} + 3\mu_3b + 2\mu_3 + 2a - 1 \right) &= i^{ 2 }
\\
\chi^A\left(((\mu_3 + 1)b - 2)c + (3a\cdot \mu_3 + 3a)\cdot b + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((2a\cdot \mu_3 + 2a)\cdot b^{2} - 2\mu_3)c + (4\mu_3 + 4)b^{2} + 4a\cdot \mu_3 + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((-\mu_3 - 1)b^{2} - 2b + 4\mu_3)c + (2a\cdot \mu_3 + 2a)\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 1 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (8a + 16 )x^{46} + (16a + 8 )x^{45} + (26a + 16 )x^{44} + (24a + 8 )x^{43} + 28 x^{42} + 4a x^{40} + 24a x^{39} + 24 x^{38} + 24a x^{37} + 8a x^{36} + (16a + 24 )x^{35} + (16a + 16 )x^{34} + 16a x^{33} + (12a + 16 )x^{32} + 16a x^{31} + (16a + 16 )x^{30} + (16a + 16 )x^{29} + (20a + 20 )x^{28} + 8 x^{27} + 12a x^{26} + (16a + 16 )x^{25} + 30a x^{24} + (24a + 16 )x^{22} + 24a x^{21} + (8a + 20 )x^{20} + (24a + 16 )x^{19} + 24a x^{18} + 24 x^{17} + (8a + 8 )x^{16} + 16 x^{15} + 16 x^{14} + 12 x^{12} + 8a x^{11} + (8a + 8 )x^{10} + 8 x^{8} + (16a + 16 )x^{6} + 16 x^{5} + (12a + 24 )x^{4} + 16a x^{3} + (16a + 16 )x^{2} + (24a + 16 )x + 14 \)