Properties

Label 13.10.8.1
Base \(\Q_{13}\)
Degree \(10\)
e \(5\)
f \(2\)
c \(8\)
Galois group $F_5$ (as 10T4)

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Defining polynomial

\(x^{10} + 60 x^{9} + 1450 x^{8} + 17760 x^{7} + 112360 x^{6} + 319418 x^{5} + 225500 x^{4} + 89240 x^{3} + 226880 x^{2} + 1274440 x + 3013497\) Copy content Toggle raw display

Invariants

Base field: $\Q_{13}$
Degree $d$: $10$
Ramification exponent $e$: $5$
Residue field degree $f$: $2$
Discriminant exponent $c$: $8$
Discriminant root field: $\Q_{13}(\sqrt{2})$
Root number: $1$
$\card{ \Aut(K/\Q_{ 13 }) }$: $2$
This field is not Galois over $\Q_{13}.$
Visible slopes:None

Intermediate fields

$\Q_{13}(\sqrt{2})$, 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:$\Q_{13}(\sqrt{2})$ $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{2} + 12 x + 2 \) 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:$2$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$F_5$ (as 10T4)
Inertia group:Intransitive group isomorphic to $C_5$
Wild inertia group:$C_1$
Unramified degree:$4$
Tame degree:$5$
Wild slopes:None
Galois mean slope:$4/5$
Galois splitting model:$x^{10} - x^{8} - 26 x^{7} - 36 x^{6} - 52 x^{5} + 54 x^{4} + 130 x^{3} - 125 x^{2} - 125$