Properties

Label 2.15.10.1
Base \(\Q_{2}\)
Degree \(15\)
e \(3\)
f \(5\)
c \(10\)
Galois group $S_3 \times C_5$ (as 15T4)

Related objects

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Defining polynomial

\( x^{15} + 8 x^{6} + 32 \)

Invariants

Base field: $\Q_{2}$
Degree $d$: $15$
Ramification exponent $e$: $3$
Residue field degree $f$: $5$
Discriminant exponent $c$: $10$
Discriminant root field: $\Q_{2}(\sqrt{5})$
Root number: $1$
$|\Aut(K/\Q_{ 2 })|$: $5$
This field is not Galois over $\Q_{2}.$

Intermediate fields

2.3.2.1, 2.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:2.5.0.1 $\cong \Q_{2}(t)$ where $t$ is a root of \( x^{5} + x^{2} + 1 \)
Relative Eisenstein polynomial:$ x^{3} - 2 t \in\Q_{2}(t)[x]$

Invariants of the Galois closure

Galois group:$C_5\times S_3$ (as 15T4)
Inertia group:Intransitive group isomorphic to $C_3$
Unramified degree:$10$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model:Not computed