Properties

Label 7.14.16.1
Base \(\Q_{7}\)
Degree \(14\)
e \(7\)
f \(2\)
c \(16\)
Galois group $C_7^2:C_6$ (as 14T14)

Related objects

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Defining polynomial

\( x^{14} + 28 x^{13} + 21 x^{12} + 42 x^{11} + 14 x^{10} + 28 x^{8} + 37 x^{7} + 28 x^{5} + 7 x^{4} + 7 x^{3} + 35 x^{2} + 28 x + 31 \)

Invariants

Base field: $\Q_{7}$
Degree $d$ : $14$
Ramification exponent $e$ : $7$
Residue field degree $f$ : $2$
Discriminant exponent $c$ : $16$
Discriminant root field: $\Q_{7}(\sqrt{*})$
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} + 7 t x^{6} + \left(21 t + 42\right) x^{5} + \left(21 t + 28\right) x^{4} + \left(21 t + 35\right) x^{3} + \left(7 t + 21\right) x^{2} + 42 t + 21 \in\Q_{7}(t)[x]$

Invariants of the Galois closure

Galois group:$C_7^2:C_6$ (as 14T14)
Inertia group:Intransitive group isomorphic to $C_7^2:C_3$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:[4/3, 4/3]
Galois mean slope:$194/147$
Galois splitting model:$x^{14} - 84 x^{12} - 84 x^{11} + 3206 x^{10} + 4956 x^{9} - 66668 x^{8} - 130154 x^{7} + 786842 x^{6} + 1801856 x^{5} - 4775288 x^{4} - 13285916 x^{3} + 9985696 x^{2} + 39242252 x + 21880010$