ex.24.8.1.33_67_101.a
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
\( v(N) = 8\)
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 \(x^{2} + ((70425033346012744527594622521a - 70425033346012744527594622521)b^{3} )x + ((-105637550019019116791391933781a - 211275100038038233582783867562)b + (-140850066692025489055189245042a + 140850066692025489055189245041))b \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
4
Conductor exponent
14
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 14 })^\times/U_{\mathfrak{p}^{ 14 } }\)
:
\(\begin{array}{l}
\chi^A\left(\mu_3c + 1 \right) &= i^{ 0 }
\\
\chi^A\left((((2a + 2)\mu_3 - 2)b^{2} + (4\mu_3 - 2)b - 2\mu_3 + 4a + 4)c + ((3a + 3)\mu_3 + 2a)\cdot b^{2} + ((a - 3)\mu_3 + (a - 3))b + (2a + 4)\mu_3 - a + 1 \right) &= i^{ 2 }
\\
\chi^A\left((((a + 4)\mu_3 + (2a + 2))b^{2} + ((2a + 4)\mu_3 + 4)b - 2a\cdot \mu_3)c + (2\mu_3 + 2a)\cdot b^{2} + (2\mu_3 + 2)b + 3a\cdot \mu_3 - a - 3 \right) &= i^{ 0 }
\\
\chi^A\left(((3a\cdot \mu_3 + (a + 3))b^{2} + ((2a - 2)\mu_3 + (3a - 1))b + (a\cdot \mu_3 + 2a))\cdot c + (\mu_3 + 3)b^{2} + ((3a + 1)\mu_3 + (2a - 1))b + (-a + 1)\mu_3 - a - 1 \right) &= i^{ 2 }
\\
\chi^A\left(((2a + 1)b^{2} + 2b)\cdot c + 2a\cdot b^{2} + 2a\cdot b + 1 \right) &= i^{ 0 }
\\
\chi^A\left((((2a + 4)\mu_3 - 3)b^{2} + ((2a + 4)\mu_3 + (a + 4))b + ((4a + 4)\mu_3 - 1))c + ((2a + 4)\mu_3 + (2a - 3))b^{2} + (4\mu_3 + 3a)\cdot b + -2a - 1 \right) &= i^{ 0 }
\\
\chi^A\left(((2a + 2)\mu_3b^{2} + ((3a + 4)\mu_3 + (2a + 4))b + ((3a + 4)\mu_3 + 4a))\cdot c + ((a + 2)\mu_3 + 4)b^{2} + ((2a + 3)\mu_3 + 4)b + (-2a - 3)\mu_3 \right) &= i^{ 0 }
\\
\chi^A\left((3a\cdot \mu_3b^{2} + (a + 4)\mu_3b + 2a\cdot \mu_3)c + (a - 2)\mu_3b^{2} + (2a + 1)\mu_3b + (-2a - 1)\mu_3 \right) &= i^{ 0 }
\\
\chi^A\left(((-2\mu_3 + a)b^{2} + (4\mu_3 + (3a + 4))b + ((-a + 2)\mu_3 - 2a + 4))c + ((a + 2)\mu_3 - 2)b^{2} + ((2a - 1)\mu_3 + 4)b + \mu_3 + 4 \right) &= i^{ 2 }
\\
\chi^A\left(((4\mu_3 + 4)b^{2} + (2a + 4)b + (4\mu_3 + (4a + 4)))c + ((a + 2)\mu_3 + (a + 4))b^{2} + (\mu_3 + (2a - 3))b + (-2a - 2)\mu_3 - 2a - 1 \right) &= i^{ 2 }
\\
\chi^A\left((((2a - 2)\mu_3 + (2a + 2))b^{2} + (3a\cdot \mu_3 + (a + 4))b + (3a\cdot \mu_3 - a + 4))c + (-\mu_3 + 3)b^{2} + ((a - 3)\mu_3 + (a + 1))b + (-a - 3)\mu_3 + 3a - 3 \right) &= i^{ 0 }
\\
\chi^A\left(((a + 2)b^{2} + (2a - 1)b + (2a - 2))c + (2a - 2)b^{2} + (3a + 2)b + 1 \right) &= i^{ 0 }
\\
\chi^A\left((((a + 2)\mu_3 - 2)b^{2} + ((2a + 4)\mu_3 + (3a + 4))b + (2\mu_3 + (3a - 2)))c + ((2a - 1)\mu_3 + (a - 2))b^{2} + ((3a - 1)\mu_3 + (a + 3))b + (-2a + 2)\mu_3 - a + 3 \right) &= i^{ 2 }
\\
\chi^A\left(((2a + 1)\mu_3b^{2} + 2\mu_3b)\cdot c + 2a\cdot \mu_3b^{2} + 2a\cdot \mu_3b + 1 \right) &= i^{ 1 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (24a + 12 )x^{47} + (18a + 4 )x^{46} + (4a + 4 )x^{45} + (4a + 24 )x^{44} + 16 x^{43} + 16 x^{42} + (12a + 20 )x^{41} + (14a + 28 )x^{40} + 20a x^{39} + (28a + 8 )x^{38} + (20a + 16 )x^{37} + (20a + 4 )x^{36} + 8a x^{35} + 16a x^{34} + 16 x^{33} + (28a + 16 )x^{32} + (12a + 4 )x^{31} + (20a + 28 )x^{30} + (28a + 8 )x^{29} + (28a + 20 )x^{28} + (8a + 24 )x^{27} + (28a + 28 )x^{26} + 4 x^{25} + (2a + 4 )x^{24} + (12a + 24 )x^{23} + (28a + 20 )x^{22} + (12a + 8 )x^{21} + (16a + 24 )x^{20} + (12a + 8 )x^{17} + (16a + 28 )x^{16} + 24 x^{15} + (16a + 16 )x^{14} + (24a + 8 )x^{13} + (28a + 12 )x^{12} + 24a x^{11} + (8a + 16 )x^{10} + (8a + 16 )x^{9} + (8a + 16 )x^{8} + (4a + 16 )x^{7} + (28a + 16 )x^{6} + (8a + 8 )x^{5} + (4a + 24 )x^{4} + (16a + 16 )x^{3} + (28a + 24 )x^{2} + (28a + 16 )x + 30a + 18 \)