Properties

Label 7.7.7.3
Base \(\Q_{7}\)
Degree \(7\)
e \(7\)
f \(1\)
c \(7\)
Galois group $F_7$ (as 7T4)

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Defining polynomial

\(x^{7} + 35 x + 7\) Copy content Toggle raw display

Invariants

Base field: $\Q_{7}$
Degree $d$: $7$
Ramification exponent $e$: $7$
Residue field degree $f$: $1$
Discriminant exponent $c$: $7$
Discriminant root field: $\Q_{7}(\sqrt{7})$
Root number: $i$
$\card{ \Aut(K/\Q_{ 7 }) }$: $1$
This field is not Galois over $\Q_{7}.$
Visible slopes:$[7/6]$

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: \( x^{7} + 35 x + 7 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 2$
Associated inertia:$1$
Indices of inseparability:$[1, 0]$

Invariants of the Galois closure

Galois group:$F_7$ (as 7T4)
Inertia group:$F_7$ (as 7T4)
Wild inertia group:$C_7$
Unramified degree:$1$
Tame degree:$6$
Wild slopes:$[7/6]$
Galois mean slope:$47/42$
Galois splitting model:$x^{7} - 21 x^{5} - 21 x^{4} + 259 x^{3} - 189 x^{2} - 126 x - 293$