Properties

Label 13.15.12.1
Base \(\Q_{13}\)
Degree \(15\)
e \(5\)
f \(3\)
c \(12\)
Galois group $F_5\times C_3$ (as 15T8)

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Defining polynomial

\(x^{15} + 10 x^{13} + 55 x^{12} + 40 x^{11} + 479 x^{10} + 1290 x^{9} + 930 x^{8} - 5530 x^{7} + 16110 x^{6} + 19349 x^{5} + 267515 x^{4} + 64685 x^{3} + 204430 x^{2} - 112095 x + 176534\) Copy content Toggle raw display

Invariants

Base field: $\Q_{13}$
Degree $d$: $15$
Ramification exponent $e$: $5$
Residue field degree $f$: $3$
Discriminant exponent $c$: $12$
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

13.3.0.1, 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:13.3.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{3} + 2 x + 11 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{5} + 13 \) $\ \in\Q_{13}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{4} + 5z^{3} + 10z^{2} + 10z + 5$
Associated inertia:$4$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_3\times F_5$ (as 15T8)
Inertia group:Intransitive group isomorphic to $C_5$
Wild inertia group:$C_1$
Unramified degree:$12$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:Not computed