Properties

Label 139.4.2.1
Base \(\Q_{139}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(2\)
Galois group $C_2^2$ (as 4T2)

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Defining polynomial

\(x^{4} + 417 x^{2} + 77284\)  Toggle raw display

Invariants

Base field: $\Q_{139}$
Degree $d$: $4$
Ramification exponent $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $2$
Discriminant root field: $\Q_{139}$
Root number: $1$
$|\Gal(K/\Q_{ 139 })|$: $4$
This field is Galois and abelian over $\Q_{139}.$

Intermediate fields

$\Q_{139}(\sqrt{2})$, $\Q_{139}(\sqrt{139})$, $\Q_{139}(\sqrt{139\cdot 2})$

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{139}(\sqrt{2})$ $\cong \Q_{139}(t)$ where $t$ is a root of \( x^{2} - x + 2 \)  Toggle raw display
Relative Eisenstein polynomial:\( x^{2} - 139 t^{2} \)$\ \in\Q_{139}(t)[x]$  Toggle raw display

Invariants of the Galois closure

Galois group:$C_2^2$ (as 4T2)
Inertia group:Intransitive group isomorphic to $C_2$
Unramified degree:$2$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model:$x^{4} + 417 x^{2} + 77284$  Toggle raw display