Properties

Label 13.15.14.1
Base \(\Q_{13}\)
Degree \(15\)
e \(15\)
f \(1\)
c \(14\)
Galois group $F_5\times C_3$ (as 15T8)

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Defining polynomial

\(x^{15} + 13\) Copy content Toggle raw display

Invariants

Base field: $\Q_{13}$
Degree $d$: $15$
Ramification exponent $e$: $15$
Residue field degree $f$: $1$
Discriminant exponent $c$: $14$
Discriminant root field: $\Q_{13}(\sqrt{2})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 13 }) }$: $3$
This field is not Galois over $\Q_{13}.$
Visible slopes:None

Intermediate fields

13.3.2.2, 13.5.4.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{13}$
Relative Eisenstein polynomial: \( x^{15} + 13 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{14} + 2z^{13} + z^{12} + z + 2$
Associated inertia:$4$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_3\times F_5$ (as 15T8)
Inertia group:$C_{15}$ (as 15T1)
Wild inertia group:$C_1$
Unramified degree:$4$
Tame degree:$15$
Wild slopes:None
Galois mean slope:$14/15$
Galois splitting model:Not computed