Properties

Label 127.6.4.2
Base \(\Q_{127}\)
Degree \(6\)
e \(3\)
f \(2\)
c \(4\)
Galois group $C_6$ (as 6T1)

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Defining polynomial

\(x^{6} - 16002 x^{3} + 48387\) Copy content Toggle raw display

Invariants

Base field: $\Q_{127}$
Degree $d$: $6$
Ramification exponent $e$: $3$
Residue field degree $f$: $2$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{127}(\sqrt{3})$
Root number: $1$
$\card{ \Gal(K/\Q_{ 127 }) }$: $6$
This field is Galois and abelian over $\Q_{127}.$
Visible slopes:None

Intermediate fields

$\Q_{127}(\sqrt{3})$, 127.3.2.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Unramified/totally ramified tower

Unramified subfield:$\Q_{127}(\sqrt{3})$ $\cong \Q_{127}(t)$ where $t$ is a root of \( x^{2} + 126 x + 3 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{3} + 127 t \) $\ \in\Q_{127}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Data not computed

Invariants of the Galois closure

Galois group: $C_6$ (as 6T1)
Inertia group: Intransitive group isomorphic to $C_3$
Wild inertia group: $C_1$
Unramified degree: $2$
Tame degree: $3$
Wild slopes: None
Galois mean slope: $2/3$
Galois splitting model: $x^{6} - x^{5} + 2 x^{4} + 4378 x^{3} + 130035 x^{2} - 721815 x + 2868703$ Copy content Toggle raw display