Base \(\Q_{7}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(2\)
Galois group $C_2^2$ (as 4T2)

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Defining polynomial

\(x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268\) Copy content Toggle raw display


Base field: $\Q_{7}$
Degree $d$: $4$
Ramification exponent $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $2$
Discriminant root field: $\Q_{7}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 7 }) }$: $4$
This field is Galois and abelian over $\Q_{7}.$
Visible slopes:None

Intermediate fields

$\Q_{7}(\sqrt{7})$, $\Q_{7}(\sqrt{7\cdot 3})$, $\Q_{7}(\sqrt{3})$

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}(\sqrt{3})$ $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{2} + 6 x + 3 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + 7 \) $\ \in\Q_{7}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 2$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_2^2$ (as 4T2)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model: $x^{4} + 35 x^{2} + 441$ Copy content Toggle raw display