Properties

Label 71.7.6.2
Base \(\Q_{71}\)
Degree \(7\)
e \(7\)
f \(1\)
c \(6\)
Galois group $C_7$ (as 7T1)

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Defining polynomial

\(x^{7} + 142\) Copy content Toggle raw display

Invariants

Base field: $\Q_{71}$
Degree $d$: $7$
Ramification exponent $e$: $7$
Residue field degree $f$: $1$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{71}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 71 }) }$: $7$
This field is Galois and abelian over $\Q_{71}.$
Visible slopes:None

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 71 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{71}$
Relative Eisenstein polynomial: \( x^{7} + 142 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{6} + 7z^{5} + 21z^{4} + 35z^{3} + 35z^{2} + 21z + 7$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_7$ (as 7T1)
Inertia group:$C_7$ (as 7T1)
Wild inertia group:$C_1$
Unramified degree:$1$
Tame degree:$7$
Wild slopes:None
Galois mean slope:$6/7$
Galois splitting model:Not computed