ex.8.6.4.21_51_97.c
Base Field
\(F = \) 2.2.1.0a1.1 \( = \mathbb{Q}_{ 2 }(a) = \mathbb{Q}_{ 2 }[x] / (x^{2} + x + 1 )\)
View on LMFDB ↗
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
\((\mathcal{O}_K/\mathfrak p^{ 4 })^\times/U_{\mathfrak{p}^{ 4 } }\)
:
\(\begin{array}{l}
\chi^A\left(b^{2}c + 1 \right) &= i^{ 1 }
\\
\chi^A\left((4a\cdot b^{2} + 4b + (4a + 4))c + (4a + 4)b^{2} - 2a\cdot b - 2a - 3 \right) &= i^{ 2 }
\\
\chi^A\left((-2a\cdot b^{2} + (2a - 1)b - 2a + 4)c + (2a + 4)b^{2} + (2a + 3)b - 2a \right) &= i^{ 2 }
\\
\chi^A\left(((a + 1)b + a)\cdot c + 1 \right) &= i^{ 2 }
\\
\chi^A\left((b^{2} + b + a)c + 1 \right) &= i^{ 1 }
\\
\chi^A\left(((4a - 3)b^{2} + (a + 3)b + (4a + 1))c + (2a - 2)b^{2} + (2a + 3)b - 2a - 2 \right) &= i^{ 1 }
\\
\chi^A\left(((a + 1)b^{2} + (a + 1)b + 1)c + 1 \right) &= i^{ 0 }
\\
\chi^A\left(((-a - 3)b^{2} + (a - 3)b - 3a + 3)c + (-3a - 1)b^{2} + (3a + 3)b - 3a + 3 \right) &= i^{ 0 }
\end{array}
\)
Inertia Polynomial
The following polynomial defines a field \(L\) such that \(L^{un}\) is the fixed field of \(\tau\).
\( x^{24} + (396a + 212 )x^{23} + (744a + 572 )x^{22} + (460a + 736 )x^{21} + (504a + 292 )x^{20} + (920a + 996 )x^{19} + (72a + 600 )x^{18} + (144a + 488 )x^{17} + (116a + 292 )x^{16} + (400a + 16 )x^{15} + (260a + 728 )x^{14} + (104a + 808 )x^{13} + (224a + 772 )x^{12} + (680a + 104 )x^{11} + (24a + 344 )x^{10} + (976a + 944 )x^{9} + (636a + 880 )x^{8} + (480a + 576 )x^{7} + (592a + 368 )x^{6} + (336a + 736 )x^{5} + (296a + 328 )x^{4} + 992a x^{3} + (96a + 432 )x^{2} + (192a + 448 )x + 656a + 24 \)