Properties

Label 7.6.3.1
Base \(\Q_{7}\)
Degree \(6\)
e \(2\)
f \(3\)
c \(3\)
Galois group $C_6$ (as 6T1)

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Defining polynomial

\(x^{6} + 861 x^{5} + 33033 x^{4} + 1385475 x^{3} + 277830 x^{2} + 8232 x - 1372\) Copy content Toggle raw display

Invariants

Base field: $\Q_{7}$
Degree $d$: $6$
Ramification exponent $e$: $2$
Residue field degree $f$: $3$
Discriminant exponent $c$: $3$
Discriminant root field: $\Q_{7}(\sqrt{7})$
Root number: $i$
$\card{ \Gal(K/\Q_{ 7 }) }$: $6$
This field is Galois and abelian over $\Q_{7}.$
Visible slopes:None

Intermediate fields

$\Q_{7}(\sqrt{7})$, 7.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:7.3.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{3} + 6 x^{2} + 4 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + \left(28 t^{2} + 42 t + 35\right) x + 7 t \) $\ \in\Q_{7}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Data not computed

Invariants of the Galois closure

Galois group: $C_6$ (as 6T1)
Inertia group: Intransitive group isomorphic to $C_2$
Wild inertia group: $C_1$
Unramified degree: $3$
Tame degree: $2$
Wild slopes: None
Galois mean slope: $1/2$
Galois splitting model:$x^{6} - x^{5} + 21 x^{4} - 22 x^{3} + 58 x^{2} + 23 x + 155$