Properties

Label 7.14.14.2
Base \(\Q_{7}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(14\)
Galois group 14T23

Related objects

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Defining polynomial

\( x^{14} + 14 x^{13} + 21 x^{11} + 14 x^{10} + 14 x^{8} + 15 x^{7} + 14 x^{6} + 21 x^{5} + 21 x^{3} + 21 x^{2} + 35 x + 45 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $14$
Ramification exponent $e$ : $7$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $14$
Discriminant root field: $\Q_{7}$
Root number: $-1$
$|\Aut(K/\Q_{ 7 })|$: $1$
This field is not Galois over $\Q_{7}$.

Intermediate fields

$\Q_{7}(\sqrt{*})$

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}(\sqrt{*})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} - x + 3 \)
Relative Eisenstein polynomial:$ x^{7} + 35 x^{6} + \left(28 t + 21\right) x^{5} + \left(28 t + 42\right) x^{4} + \left(7 t + 42\right) x^{3} + 28 x^{2} + \left(28 t + 21\right) x + 14 t + 14 \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:14T23
Inertia group:Intransitive group isomorphic to $C_7^2:C_3:C_2$
Unramified degree:$2$
Tame degree:$6$
Wild slopes:[7/6, 7/6]
Galois mean slope:$341/294$
Galois splitting model:$x^{14} - 14 x^{12} + 77 x^{10} - 210 x^{8} - 11 x^{7} + 294 x^{6} + 77 x^{5} - 196 x^{4} - 154 x^{3} + 49 x^{2} + 77 x + 29$