ex.8.5.4.35_67_95.c
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
\( v(N) = 5\)
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 \(x^{2} + 2 x + ((-12315268912454247165685184694a - 140859127093795256702533651142)b^{2} + (143657191841227645937611462065a - 113388180470743875764844059666)b + (43780640976257799590642092457a + 20888447497506072635599363512))\cdot 2 \)
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
\((\mathcal{O}_K/\mathfrak p^{ 3 })^\times/U_{\mathfrak{p}^{ 3 } }\)
:
\(\begin{array}{l}
\chi^A\left((b + 1)c + 1 \right) &= i^{ 2 }
\\
\chi^A\left((2a\cdot b^{2} + (2a + 4)b + 1)c + 2a\cdot b^{2} + (-2a + 4)b - 3 \right) &= i^{ 2 }
\\
\chi^A\left((4a\cdot b + 4)c + (4a + 3)b^{2} + (2a - 2)b + 4 \right) &= i^{ 2 }
\\
\chi^A\left(((-2a + 1)b^{2} + (-a + 3)b + (4a + 1))c - 2b^{2} + (2a - 1)b + 4a + 4 \right) &= i^{ 1 }
\\
\chi^A\left(((a + 1)b + a)\cdot c + 1 \right) &= i^{ 0 }
\\
\chi^A\left(b^{2}c + 1 \right) &= i^{ 0 }
\\
\chi^A\left((b^{2} + b + a)c + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((2a - 2)b^{2} + (4a + 2)b + (4a + 2))c + 2a\cdot b^{2} + (-2a + 4)b + 2a - 1 \right) &= i^{ 2 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{24} + (264a + 528 )x^{23} + (580a + 948 )x^{22} + (772a + 424 )x^{21} + (600a + 672 )x^{20} + (208a + 192 )x^{19} + (972a + 372 )x^{18} + (776a + 296 )x^{17} + (848a + 200 )x^{16} + (152a + 784 )x^{15} + (224a + 752 )x^{14} + (960a + 864 )x^{13} + (216a + 448 )x^{12} + (832a + 192 )x^{11} + (912a + 672 )x^{10} + (584a + 88 )x^{9} + (544a + 688 )x^{8} + (992a + 416 )x^{7} + (392a + 176 )x^{6} + (352a + 128 )x^{5} + (784a + 272 )x^{4} + (448a + 320 )x^{3} + (128a + 352 )x^{2} + (160a + 864 )x + 120a + 296 \)