ex.8.6.4.21_51_97.a
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 \(x^{2} + 2 x + ((-49861771308763385703883439913a - 156250771962343225720271213443)b^{2} + (10033211722547260229267252575a - 4731761391742728921849352311)b + (92361363490280408299853204666a - 53660844451361984880706855254))\cdot 2 \)
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
\((\mathcal{O}_K/\mathfrak p^{ 4 })^\times/U_{\mathfrak{p}^{ 4 } }\)
:
\(\begin{array}{l}
\chi^A\left(b^{2}c + 1 \right) &= i^{ 1 }
\\
\chi^A\left((4a\cdot b^{2} + 4b + (4a + 4))c + (4a + 4)b^{2} - 2a\cdot b - 2a - 3 \right) &= i^{ 2 }
\\
\chi^A\left((-2a\cdot b^{2} + (2a - 1)b - 2a + 4)c + (2a + 4)b^{2} + (2a + 3)b - 2a \right) &= i^{ 0 }
\\
\chi^A\left(((a + 1)b + a)\cdot c + 1 \right) &= i^{ 2 }
\\
\chi^A\left((b^{2} + b + a)c + 1 \right) &= i^{ 1 }
\\
\chi^A\left(((4a - 3)b^{2} + (a + 3)b + (4a + 1))c + (2a - 2)b^{2} + (2a + 3)b - 2a - 2 \right) &= i^{ 1 }
\\
\chi^A\left(((a + 1)b^{2} + (a + 1)b + 1)c + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((-a - 3)b^{2} + (a - 3)b - 3a + 3)c + (-3a - 1)b^{2} + (3a + 3)b - 3a + 3 \right) &= i^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{24} + (836a + 444 )x^{23} + (36a + 450 )x^{22} + (552a + 56 )x^{21} + (160a + 34 )x^{20} + (308a + 284 )x^{19} + (336a + 792 )x^{18} + (760a + 488 )x^{17} + (164a + 358 )x^{16} + (560a + 656 )x^{15} + (320a + 1008 )x^{14} + (864a + 800 )x^{13} + (256a + 360 )x^{12} + (800a + 928 )x^{11} + (64a + 976 )x^{10} + (288a + 416 )x^{9} + (48a + 372 )x^{8} + (976a + 304 )x^{7} + (1008a + 936 )x^{6} + (160a + 608 )x^{5} + 56 x^{4} + (16a + 368 )x^{3} + (128a + 544 )x^{2} + (32a + 224 )x + 880a + 984 \)