Properties

Label 61.9.6.1
Base \(\Q_{61}\)
Degree \(9\)
e \(3\)
f \(3\)
c \(6\)
Galois group $C_3^2$ (as 9T2)

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Defining polynomial

\(x^{9} + 21 x^{7} + 360 x^{6} + 147 x^{5} + 1197 x^{4} - 53630 x^{3} + 17640 x^{2} - 158760 x + 1748923\) Copy content Toggle raw display

Invariants

Base field: $\Q_{61}$
Degree $d$: $9$
Ramification exponent $e$: $3$
Residue field degree $f$: $3$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{61}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 61 }) }$: $9$
This field is Galois and abelian over $\Q_{61}.$
Visible slopes:None

Intermediate fields

61.3.0.1, 61.3.2.1, 61.3.2.2, 61.3.2.3

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

Unramified/totally ramified tower

Unramified subfield:61.3.0.1 $\cong \Q_{61}(t)$ where $t$ is a root of \( x^{3} + 7 x + 59 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{3} + 61 \) $\ \in\Q_{61}(t)[x]$ 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