Properties

Label 7.11.0.1
Base \(\Q_{7}\)
Degree \(11\)
e \(1\)
f \(11\)
c \(0\)
Galois group $C_{11}$ (as 11T1)

Related objects

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Defining polynomial

\(x^{11} - 2 x + 4\)  Toggle raw display

Invariants

Base field: $\Q_{7}$
Degree $d$: $11$
Ramification exponent $e$: $1$
Residue field degree $f$: $11$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{7}$
Root number: $1$
$|\Gal(K/\Q_{ 7 })|$: $11$
This field is Galois and abelian over $\Q_{7}.$

Intermediate fields

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

Unramified/totally ramified tower

Unramified subfield:7.11.0.1 $\cong \Q_{7}(t)$ where $t$ is a root of \( x^{11} - 2 x + 4 \)  Toggle raw display
Relative Eisenstein polynomial:\( x - 7 \)$\ \in\Q_{7}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_{11}$ (as 11T1)
Inertia group:trivial
Wild inertia group:$C_1$
Unramified degree:$11$
Tame degree:$1$
Wild slopes:None
Galois mean slope:$0$
Galois splitting model:$x^{11} - x^{10} - 10 x^{9} + 9 x^{8} + 36 x^{7} - 28 x^{6} - 56 x^{5} + 35 x^{4} + 35 x^{3} - 15 x^{2} - 6 x + 1$