Properties

Label 13.8.7.3
Base \(\Q_{13}\)
Degree \(8\)
e \(8\)
f \(1\)
c \(7\)
Galois group $C_8:C_2$ (as 8T7)

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Defining polynomial

\(x^{8} + 26\) Copy content Toggle raw display

Invariants

Base field: $\Q_{13}$
Degree $d$: $8$
Ramification exponent $e$: $8$
Residue field degree $f$: $1$
Discriminant exponent $c$: $7$
Discriminant root field: $\Q_{13}(\sqrt{13\cdot 2})$
Root number: $-1$
$\card{ \Aut(K/\Q_{ 13 }) }$: $4$
This field is not Galois over $\Q_{13}.$
Visible slopes:None

Intermediate fields

$\Q_{13}(\sqrt{13\cdot 2})$, 13.4.3.3

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: \( x^{8} + 26 \) Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{7} + 8z^{6} + 2z^{5} + 4z^{4} + 5z^{3} + 4z^{2} + 2z + 8$
Associated inertia:$2$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$\OD_{16}$ (as 8T7)
Inertia group:$C_8$ (as 8T1)
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$8$
Wild slopes:None
Galois mean slope:$7/8$
Galois splitting model:$x^{8} - 3 x^{7} + 8 x^{6} - 71 x^{5} + 275 x^{4} - 466 x^{3} + 648 x^{2} - 608 x + 256$