Base \(\Q_{79}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(2\)
Galois group $C_2^2$ (as 4T2)

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Defining polynomial

\(x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412\) Copy content Toggle raw display


Base field: $\Q_{79}$
Degree $d$: $4$
Ramification exponent $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $2$
Discriminant root field: $\Q_{79}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 79 }) }$: $4$
This field is Galois and abelian over $\Q_{79}.$
Visible slopes:None

Intermediate fields

$\Q_{79}(\sqrt{3})$, $\Q_{79}(\sqrt{79})$, $\Q_{79}(\sqrt{79\cdot 3})$

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{79}(\sqrt{3})$ $\cong \Q_{79}(t)$ where $t$ is a root of \( x^{2} + 78 x + 3 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + 79 \) $\ \in\Q_{79}(t)[x]$ 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:$C_2^2$ (as 4T2)
Inertia group:Intransitive group isomorphic to $C_2$
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$2$
Wild slopes:None
Galois mean slope:$1/2$
Galois splitting model: $x^{4} + 395 x^{2} + 56169$ Copy content Toggle raw display