Properties

Label 2.14.14.7
Base \(\Q_{2}\)
Degree \(14\)
e \(2\)
f \(7\)
c \(14\)
Galois group $C_2 \wr C_7$ (as 14T29)

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Defining polynomial

\(x^{14} + 2 x^{13} + 42 x^{12} + 704 x^{11} + 3660 x^{10} + 11000 x^{9} + 40152 x^{8} + 158272 x^{7} + 362736 x^{6} + 302048 x^{5} - 214176 x^{4} - 417536 x^{3} - 79552 x^{2} + 172160 x + 89728\) Copy content Toggle raw display

Invariants

Base field: $\Q_{2}$
Degree $d$: $14$
Ramification exponent $e$: $2$
Residue field degree $f$: $7$
Discriminant exponent $c$: $14$
Discriminant root field: $\Q_{2}(\sqrt{-5})$
Root number: $-i$
$\card{ \Aut(K/\Q_{ 2 }) }$: $2$
This field is not Galois over $\Q_{2}.$
Visible slopes:$[2]$

Intermediate fields

2.7.0.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:2.7.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{7} + x + 1 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + \left(2 t^{6} + 2 t^{5} + 2 t^{4} + 2 t^{2} + 2\right) x + 4 t^{6} + 4 t^{2} + 4 t + 2 \) $\ \in\Q_{2}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + t^{6} + t^{5} + t^{4} + t^{2} + 1$
Associated inertia:$1$
Indices of inseparability:$[1, 0]$

Invariants of the Galois closure

Galois group:$C_2\wr C_7$ (as 14T29)
Inertia group:Intransitive group isomorphic to $C_2^7$
Wild inertia group:$C_2^7$
Unramified degree:$7$
Tame degree:$1$
Wild slopes:$[2, 2, 2, 2, 2, 2, 2]$
Galois mean slope:$127/64$
Galois splitting model: $x^{14} + 42 x^{12} + 378 x^{10} - 3213 x^{8} - 39690 x^{6} + 88452 x^{4} + 908334 x^{2} - 2101707$ Copy content Toggle raw display