Properties

Label 31.9.6.1
Base \(\Q_{31}\)
Degree \(9\)
e \(3\)
f \(3\)
c \(6\)
Galois group $C_3^2$ (as 9T2)

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Defining polynomial

\(x^{9} + 3 x^{7} + 177 x^{6} + 3 x^{5} + 75 x^{4} - 12992 x^{3} + 177 x^{2} - 6018 x + 205410\) Copy content Toggle raw display

Invariants

Base field: $\Q_{31}$
Degree $d$: $9$
Ramification exponent $e$: $3$
Residue field degree $f$: $3$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{31}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 31 }) }$: $9$
This field is Galois and abelian over $\Q_{31}.$
Visible slopes:None

Intermediate fields

31.3.0.1, 31.3.2.1, 31.3.2.2, 31.3.2.3

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: \( x^{3} + 31 \) $\ \in\Q_{31}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{2} + 3z + 3$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_3^2$ (as 9T2)
Inertia group:Intransitive group isomorphic to $C_3$
Wild inertia group:$C_1$
Unramified degree:$3$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:Not computed