Properties

Label 137.11.0.1
Base \(\Q_{137}\)
Degree \(11\)
e \(1\)
f \(11\)
c \(0\)
Galois group $C_{11}$ (as 11T1)

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Defining polynomial

\(x^{11} + x + 134\) Copy content Toggle raw display

Invariants

Base field: $\Q_{137}$
Degree $d$: $11$
Ramification exponent $e$: $1$
Residue field degree $f$: $11$
Discriminant exponent $c$: $0$
Discriminant root field: $\Q_{137}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 137 }) }$: $11$
This field is Galois and abelian over $\Q_{137}.$
Visible slopes:None

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q_{ 137 }$.

Unramified/totally ramified tower

Unramified subfield:137.11.0.1 $\cong \Q_{137}(t)$ where $t$ is a root of \( x^{11} + x + 134 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x - 137 \) $\ \in\Q_{137}(t)[x]$ Copy content Toggle raw display

Ramification polygon

The ramification polygon is trivial for unramified extensions.

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:$x^{11} - x^{10} - 30 x^{9} + 63 x^{8} + 220 x^{7} - 698 x^{6} - 101 x^{5} + 1960 x^{4} - 1758 x^{3} + 35 x^{2} + 243 x + 29$