Properties

Label 67.4.2.1
Base \(\Q_{67}\)
Degree \(4\)
e \(2\)
f \(2\)
c \(2\)
Galois group $C_2^2$ (as 4T2)

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Defining polynomial

\(x^{4} + 126 x^{3} + 4107 x^{2} + 8694 x + 270148\) Copy content Toggle raw display

Invariants

Base field: $\Q_{67}$
Degree $d$: $4$
Ramification exponent $e$: $2$
Residue field degree $f$: $2$
Discriminant exponent $c$: $2$
Discriminant root field: $\Q_{67}$
Root number: $1$
$\card{ \Gal(K/\Q_{ 67 }) }$: $4$
This field is Galois and abelian over $\Q_{67}.$
Visible slopes:None

Intermediate fields

$\Q_{67}(\sqrt{2})$, $\Q_{67}(\sqrt{67})$, $\Q_{67}(\sqrt{67\cdot 2})$

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{67}(\sqrt{2})$ $\cong \Q_{67}(t)$ where $t$ is a root of \( x^{2} + 63 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{2} + 67 \) $\ \in\Q_{67}(t)[x]$ 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: $x^{4} + 1541 x^{2} + 646416$ Copy content Toggle raw display