Properties

Label 7.7.12.3
Base \(\Q_{7}\)
Degree \(7\)
e \(7\)
f \(1\)
c \(12\)
Galois group $C_7$ (as 7T1)

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Defining polynomial

\(x^{7} + 42 x^{6} + 105\) Copy content Toggle raw display

Invariants

Base field: $\Q_{7}$
Degree $d$: $7$
Ramification exponent $e$: $7$
Residue field degree $f$: $1$
Discriminant exponent $c$: $12$
Discriminant root field: $\Q_{7}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 7 }) }$: $7$
This field is Galois and abelian over $\Q_{7}.$
Visible slopes:$[2]$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 7 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{7}$
Relative Eisenstein polynomial: \( x^{7} + 42 x^{6} + 105 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{6} + 6$
Associated inertia:$1$
Indices of inseparability:$[6, 0]$

Invariants of the Galois closure

Galois group:$C_7$ (as 7T1)
Inertia group:$C_7$ (as 7T1)
Wild inertia group:$C_7$
Unramified degree:$1$
Tame degree:$1$
Wild slopes:$[2]$
Galois mean slope:$12/7$
Galois splitting model: $x^{7} - 321069 x^{5} - 19799255 x^{4} + 34199092627 x^{3} + 3786165513760 x^{2} - 1195116149425212 x - 176222689666434431$ Copy content Toggle raw display