Properties

Label 31.6.3.1
Base \(\Q_{31}\)
Degree \(6\)
e \(2\)
f \(3\)
c \(3\)
Galois group $C_6$ (as 6T1)

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Defining polynomial

\(x^{6} + 961 x^{2} - 834148\) Copy content Toggle raw display

Invariants

Base field: $\Q_{31}$
Degree $d$: $6$
Ramification exponent $e$: $2$
Residue field degree $f$: $3$
Discriminant exponent $c$: $3$
Discriminant root field: $\Q_{31}(\sqrt{31})$
Root number: $i$
$\card{ \Gal(K/\Q_{ 31 }) }$: $6$
This field is Galois and abelian over $\Q_{31}.$
Visible slopes:None

Intermediate fields

$\Q_{31}(\sqrt{31})$, 31.3.0.1

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

Unramified/totally ramified tower

Unramified subfield:31.3.0.1 $\cong \Q_{31}(t)$ where $t$ is a root of \( x^{3} + x + 28 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + 31 t \) $\ \in\Q_{31}(t)[x]$ 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:$C_6$ (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:$x^{6} - x^{5} - 55 x^{4} + 55 x^{3} + 841 x^{2} - 841 x - 2743$