Properties

Label 13.8.4.1
Base \(\Q_{13}\)
Degree \(8\)
e \(2\)
f \(4\)
c \(4\)
Galois group $C_4\times C_2$ (as 8T2)

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Defining polynomial

\(x^{8} + 520 x^{7} + 101458 x^{6} + 8810644 x^{5} + 288610205 x^{4} + 142111548 x^{3} + 982314112 x^{2} + 3617879976 x + 920156436\) Copy content Toggle raw display

Invariants

Base field: $\Q_{13}$
Degree $d$: $8$
Ramification exponent $e$: $2$
Residue field degree $f$: $4$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{13}$
Root number: $-1$
$\card{ \Gal(K/\Q_{ 13 }) }$: $8$
This field is Galois and abelian over $\Q_{13}.$
Visible slopes:None

Intermediate fields

$\Q_{13}(\sqrt{2})$, $\Q_{13}(\sqrt{13})$, $\Q_{13}(\sqrt{13\cdot 2})$, 13.4.0.1, 13.4.2.1, 13.4.2.2

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

Unramified/totally ramified tower

Unramified subfield:13.4.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{4} + 3 x^{2} + 12 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + 130 x + 13 \) $\ \in\Q_{13}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + 2$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_2\times C_4$ (as 8T2)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_1$
Unramified degree:$4$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{8} - x^{7} + 4 x^{6} - 7 x^{5} + 19 x^{4} + 21 x^{3} + 36 x^{2} + 27 x + 81$