Properties

Label 13.10.5.2
Base \(\Q_{13}\)
Degree \(10\)
e \(2\)
f \(5\)
c \(5\)
Galois group $C_{10}$ (as 10T1)

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Defining polynomial

\( x^{10} - 57122 x^{2} + 2227758 \)

Invariants

Base field: $\Q_{13}$
Degree $d$ : $10$
Ramification exponent $e$ : $2$
Residue field degree $f$ : $5$
Discriminant exponent $c$ : $5$
Discriminant root field: $\Q_{13}(\sqrt{13*})$
Root number: $-1$
$|\Gal(K/\Q_{ 13 })|$: $10$
This field is Galois and abelian over $\Q_{13}$.

Intermediate fields

$\Q_{13}(\sqrt{13*})$, 13.5.0.1

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

Unramified/totally ramified tower

Unramified subfield:13.5.0.1 $\cong \Q_{13}(t)$ where $t$ is a root of \( x^{5} - 2 x + 6 \)
Relative Eisenstein polynomial:$ x^{2} - 13 t \in\Q_{13}(t)[x]$

Invariants of the Galois closure

Galois group:$C_{10}$ (as 10T1)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$5$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{10} - x^{9} + 34 x^{8} - 34 x^{7} + 430 x^{6} - 430 x^{5} + 2509 x^{4} - 2509 x^{3} + 6964 x^{2} - 6964 x + 9637$