Properties

Label 3.2.3.6a1.1
Base \(\Q_{3}\)
Degree \(6\)
e \(3\)
f \(2\)
c \(6\)
Galois group $D_{6}$ (as 6T3)

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Error: Incorrect local algebra for Q(zeta3)

Defining polynomial

$( x^{2} + 2 x + 2 )^{3} + 6 ( x^{2} + 2 x + 2 ) + 3$ Copy content Toggle raw display

Invariants

Base field: $\Q_{3}$
Degree $d$: $6$
Ramification index $e$: $3$
Residue field degree $f$: $2$
Discriminant exponent $c$: $6$
Discriminant root field: $\Q_{3}(\sqrt{2})$
Root number: $1$
$\Aut(K/\Q_{3})$: $C_2$
This field is not Galois over $\Q_{3}.$
Visible Artin slopes:$[\frac{3}{2}]$
Visible Swan slopes:$[\frac{1}{2}]$
Means:$\langle\frac{1}{3}\rangle$
Rams:$(\frac{1}{2})$
Jump set:undefined
Roots of unity:$8 = (3^{ 2 } - 1)$

Intermediate fields

$\Q_{3}(\sqrt{2})$, 3.1.3.3a1.2

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

Canonical tower

Unramified subfield:$\Q_{3}(\sqrt{2})$ $\cong \Q_{3}(t)$ where $t$ is a root of \( x^{2} + 2 x + 2 \) Copy content Toggle raw display
Relative Eisenstein polynomial: \( x^{3} + 6 x + 3 \) $\ \in\Q_{3}(t)[x]$ Copy content Toggle raw display

Ramification polygon

Residual polynomials:$z + (2 t + 2)$
Associated inertia:$1$
Indices of inseparability:$[1, 0]$

Invariants of the Galois closure

Galois degree: $12$
Galois group: $D_6$ (as 6T3)
Inertia group: Intransitive group isomorphic to $S_3$
Wild inertia group: $C_3$
Galois unramified degree: $2$
Galois tame degree: $2$
Galois Artin slopes: $[\frac{3}{2}]$
Galois Swan slopes: $[\frac{1}{2}]$
Galois mean slope: $1.1666666666666667$
Galois splitting model:$x^{6} - 3 x^{5} + 21 x^{4} - 31 x^{3} + 63 x^{2} - 45 x + 50$