ex.8.6.4.53_79_101.d
Base Field
\(F = \) 2.2.1.0a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + x + 1 )\)
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Description
exceptional, Q8
Construction
Extension to \(I_F\) of \(
\tau = \operatorname{Ind}^{I_K}_{I_L} \chi
\), with \(L, K\) and \(\chi\) as below
Semistability defect
\( e = 8\)
Conductor exponent
\( v(N) = 6\)
Character Order
4
Triply Field
The inertial type \(\tau\) becomes an induction (triply imprimitive) over:
\( L = F(b) \), \(b\) a root of \(x^{3} + x^{2} + x + a \)
Inducing Field
The inertial type \(\tau\) becomes reducible over
\(K = L(c)\), \(c\) a root of \(x^{2} + 2 x + ((65559432672197504600025981774a - 125635129237176042733809576915)b^{2} + (149863625134951489728624886426a - 86154930196556564508839780306)b - 139467414102321902139900244832a + 140678413560127364501900748487)\cdot 2 \)
Underlying Character
Character \(\chi^A:\mathcal O_K^\times \to \mathbb C^\times\) with the following properties:
Order
4
Conductor exponent
4
Values on generators of
\((\mathcal{O}_K/\mathfrak p^{ 4 })^\times/U_{\mathfrak{p}^{ 4 } }\)
:
\(\begin{array}{l}
\chi^A\left(b^{2}c + 1 \right) &= i^{ 2 }
\\
\chi^A\left(c + 1 \right) &= i^{ 2 }
\\
\chi^A\left((4a\cdot b^{2} + 4b + (4a + 4))c + (4a + 2)b^{2} + (4a + 2)b + 4a + 1 \right) &= i^{ 2 }
\\
\chi^A\left(((4a - 3)b^{2} + (3a - 3)b - 2a - 3)c + (-2a + 4)b^{2} + (2a - 1)b + 2 \right) &= i^{ 2 }
\\
\chi^A\left(((a + 1)b + a)\cdot c + 1 \right) &= i^{ 1 }
\\
\chi^A\left((b^{2} + b + a)c + 1 \right) &= i^{ 1 }
\\
\chi^A\left((a\cdot b^{2} + a)c + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((4a - 2)b^{2} + (-2a - 2)b + (4a + 4))c + (2a + 4)b^{2} - 2a\cdot b - 2a + 1 \right) &= i^{ 2 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{24} + (980a + 512 )x^{23} + (332a + 166 )x^{22} + (232a + 220 )x^{21} + (212a + 40 )x^{20} + (708a + 132 )x^{19} + (692a + 732 )x^{18} + 488a x^{17} + (870a + 768 )x^{16} + 288a x^{15} + (616a + 736 )x^{14} + (16a + 704 )x^{13} + (520a + 304 )x^{12} + (928a + 816 )x^{11} + (832a + 336 )x^{10} + (224a + 128 )x^{9} + (596a + 52 )x^{8} + (64a + 976 )x^{7} + (200a + 824 )x^{6} + (912a + 368 )x^{5} + 560a x^{4} + (304a + 256 )x^{3} + (304a + 640 )x^{2} + (896a + 224 )x + 128a + 200 \)