ex.24.3.1.1_3_5.a
Base Field
\(F = \) 2.1.2.2a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + 2 x + 2 )\)
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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
\( v(N) = 3\)
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 \(x^{2} + ((a + 1)b )x + (a + 1)b \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
4
Conductor exponent
3
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 3 })^\times/U_{\mathfrak{p}^{ 3 } }\)
:
\(\begin{array}{l}
\chi^A\left(\mu_3c + 1 \right) &= i^{ 1 }
\\
\chi^A\left((((2a - 2)\mu_3 + (2a - 2))b^{2} + ((2a + 2)\mu_3 + 4a)\cdot b + ((2a + 2)\mu_3 + (4a + 4)))c + ((-a + 1)\mu_3 - a)b^{2} + ((-3a - 2)\mu_3 + 4a)\cdot b + 3a\cdot \mu_3 - 1 \right) &= i^{ 0 }
\\
\chi^A\left((((3a - 2)\mu_3 + 2)b^{2} + ((-2a + 2)\mu_3 + (2a + 2))b + ((2a + 4)\mu_3 - 2a + 4))c + ((-2a + 2)\mu_3 - 3a - 3)b^{2} + ((-a + 1)\mu_3 + (a + 2))b + (a - 3)\mu_3 + 2a + 2 \right) &= i^{ 0 }
\\
\chi^A\left((((-2a - 2)\mu_3 - a - 3)b^{2} + (a\cdot \mu_3 - a + 2)b + ((-a + 2)\mu_3 - 3a + 4))c + (2a\cdot \mu_3 - 3)b^{2} + (3\mu_3 + (2a - 3))b + (-3a + 3)\mu_3 + a \right) &= i^{ 2 }
\\
\chi^A\left((((-a + 2)\mu_3 + 4a)\cdot b^{2} + ((3a + 2)\mu_3 + 4a)\cdot b + ((2a + 4)\mu_3 + (4a - 2)))c + ((a + 2)\mu_3 - 3a + 3)b^{2} + ((-2a + 3)\mu_3 + (2a - 1))b + (4a + 4)\mu_3 - 3a - 1 \right) &= i^{ 2 }
\\
\chi^A\left((4b^{2} - \mu_3 + 3)c + 4b^{2} - \mu_3 + 4a + 3 \right) &= i^{ 0 }
\\
\chi^A\left(((-2\mu_3 - 3a + 4)b^{2} + ((-2a + 2)\mu_3 + 4a)\cdot b + ((2a + 4)\mu_3 + (4a + 4)))c + (-\mu_3 + 2)b^{2} + (a\cdot \mu_3 + 4)b + (-3a - 2)\mu_3 + 3 \right) &= i^{ 0 }
\\
\chi^A\left((((2a + 2)\mu_3 + (2a + 4))b^{2} + ((2a + 2)\mu_3 + 4)b + ((-2a + 4)\mu_3 - 2a + 2))c + ((a + 2)\mu_3 + (4a + 3))b^{2} + ((-2a + 3)\mu_3 + (3a + 3))b + (3a - 1)\mu_3 + a + 4 \right) &= i^{ 2 }
\\
\chi^A\left(((4a\cdot \mu_3 - 2a - 2)b^{2} + (2a\cdot \mu_3 + (2a + 2))b + ((-2a - 2)\mu_3 + 4))c + ((-3a + 1)\mu_3 - a + 3)b^{2} + ((-2a - 1)\mu_3 + 1)b - 2a\cdot \mu_3 + 4a - 1 \right) &= i^{ 2 }
\\
\chi^A\left((4a\cdot \mu_3b^{2} + (4\mu_3 + 4)b + (4a\cdot \mu_3 + 4a))\cdot c + ((a + 1)\mu_3 + (3a + 3))b^{2} + ((2a - 1)\mu_3 - 3)b + 2a\cdot \mu_3 + 4a - 1 \right) &= i^{ 2 }
\\
\chi^A\left((((2a - 2)\mu_3 - 2a - 2)b^{2} + ((-2a + 4)\mu_3 - 2)b + ((2a + 2)\mu_3 - 2a))\cdot c + ((-3a + 1)\mu_3 + (3a + 3))b^{2} + ((-2a - 1)\mu_3 + (4a - 3))b + (-2a + 4)\mu_3 - 1 \right) &= i^{ 2 }
\\
\chi^A\left((((-2a - 1)\mu_3 - a + 2)b^{2} + ((-a - 2)\mu_3 - a + 4)b + (2\mu_3 - 3a + 2))c + ((3a + 4)\mu_3 - a - 3)b^{2} + ((3a + 1)\mu_3 + (3a - 1))b + (3a - 1)\mu_3 + 3a + 2 \right) &= i^{ 2 }
\\
\chi^A\left(((-a\cdot \mu_3 + (3a - 2))b^{2} - 2\mu_3b + 2a\cdot \mu_3 + 4a - 2)c + ((-a + 4)\mu_3 - a + 4)b^{2} + ((4a + 3)\mu_3 + (4a + 2))b + (2a + 1)\mu_3 + a + 3 \right) &= i^{ 2 }
\\
\chi^A\left(((4a + 2)b^{2} + (-a - 2)b + (-2a - 2))c + (2a - 2)b^{2} + 4b + 4a - 3 \right) &= i^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{48} + (4a + 6 )x^{47} + (16a + 14 )x^{46} + (23a + 10 )x^{45} + (6a + 12 )x^{44} + (24a + 20 )x^{43} + (18a + 6 )x^{42} + (24a + 20 )x^{41} + (13a + 4 )x^{40} + (14a + 24 )x^{39} + (24a + 22 )x^{38} + (6a + 20 )x^{37} + (9a + 4 )x^{36} + (12a + 26 )x^{35} + (2a + 4 )x^{34} + (6a + 26 )x^{33} + (6a + 14 )x^{32} + (8a + 12 )x^{31} + (20a + 16 )x^{30} + 6 x^{29} + (28a + 10 )x^{28} + 22a x^{27} + (14a + 26 )x^{26} + (4a + 6 )x^{25} + (16a + 22 )x^{24} + 24 x^{23} + 22 x^{22} + (18a + 8 )x^{21} + (26a + 8 )x^{20} + (10a + 8 )x^{19} + 20 x^{18} + (28a + 20 )x^{17} + (18a + 26 )x^{16} + (4a + 28 )x^{15} + (12a + 8 )x^{13} + (6a + 16 )x^{12} + (24a + 16 )x^{11} + (16a + 4 )x^{10} + (24a + 8 )x^{9} + (26a + 12 )x^{8} + (12a + 8 )x^{7} + (26a + 16 )x^{6} + 28a x^{5} + (14a + 24 )x^{4} + 18a x^{3} + (28a + 28 )x^{2} + (20a + 8 )x + 10a + 6 \)