Properties

Label 11.7.6.1
Base \(\Q_{11}\)
Degree \(7\)
e \(7\)
f \(1\)
c \(6\)
Galois group $C_7:C_3$ (as 7T3)

Related objects

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Defining polynomial

\( x^{7} - 11 \)

Invariants

Base field: $\Q_{11}$
Degree $d$ : $7$
Ramification exponent $e$ : $7$
Residue field degree $f$ : $1$
Discriminant exponent $c$ : $6$
Discriminant root field: $\Q_{11}$
Root number: $1$
$|\Aut(K/\Q_{ 11 })|$: $1$
This field is not Galois over $\Q_{11}$.

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 11 }$.

Unramified/totally ramified tower

Unramified subfield:$\Q_{11}$
Relative Eisenstein polynomial:\( x^{7} - 11 \)

Invariants of the Galois closure

Galois group:$C_7:C_3$ (as 7T3)
Inertia group:$C_7$
Unramified degree:$3$
Tame degree:$7$
Wild slopes:None
Galois mean slope:$6/7$
Galois splitting model:$x^{7} - 3 x^{6} - 15 x^{5} + 39 x^{4} + 43 x^{3} - 133 x^{2} + 63 x - 7$