Properties

Label 43.6.4.1
Base \(\Q_{43}\)
Degree \(6\)
e \(3\)
f \(2\)
c \(4\)
Galois group $C_6$ (as 6T1)

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Defining polynomial

\(x^{6} + 126 x^{5} + 5301 x^{4} + 74930 x^{3} + 21321 x^{2} + 227916 x + 3171406\) Copy content Toggle raw display

Invariants

Base field: $\Q_{43}$
Degree $d$: $6$
Ramification exponent $e$: $3$
Residue field degree $f$: $2$
Discriminant exponent $c$: $4$
Discriminant root field: $\Q_{43}(\sqrt{2})$
Root number: $1$
$\card{ \Gal(K/\Q_{ 43 }) }$: $6$
This field is Galois and abelian over $\Q_{43}.$
Visible slopes:None

Intermediate fields

$\Q_{43}(\sqrt{2})$, 43.3.2.1

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

Unramified/totally ramified tower

Unramified subfield:$\Q_{43}(\sqrt{2})$ $\cong \Q_{43}(t)$ where $t$ is a root of \( x^{2} + 42 x + 3 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{3} + 43 \) $\ \in\Q_{43}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z^{2} + 3z + 3$
Associated inertia:$1$
Indices of inseparability:$[0]$

Invariants of the Galois closure

Galois group:$C_6$ (as 6T1)
Inertia group:Intransitive group isomorphic to $C_3$
Wild inertia group:$C_1$
Unramified degree:$2$
Tame degree:$3$
Wild slopes:None
Galois mean slope:$2/3$
Galois splitting model: $x^{6} - x^{5} + 218 x^{4} + 296 x^{3} + 11584 x^{2} + 7168 x + 32768$ Copy content Toggle raw display