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Properties

Label 3.2.1.0a1.1-1.3.5a
Base 3.2.1.0a1.1
Degree \(3\)
e \(3\)
f \(1\)
c \(5\)

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Defining polynomial

$x^{3} + 9b_{4} x + 3$

Invariants

Residue field characteristic: $3$
Degree: $3$
Base field: $\Q_{3}(\sqrt{2})$
Ramification index $e$: $3$
Residue field degree $f$: $1$
Discriminant exponent $c$: $5$
Absolute Artin slopes: $[\frac{5}{2}]$
Swan slopes: $[\frac{3}{2}]$
Means: $\langle1\rangle$
Rams: $(\frac{3}{2})$
Field count: $6$ (complete)
Ambiguity: $1$
Mass: $9$
Absolute Mass: $9/2$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 3 }$ within this relative family, not the relative extension.

Galois group: $D_{6}$ (show 3), $S_3^2$ (show 3)
Hidden Artin slopes: $[\ ]_{2}$ (show 3), $[\frac{3}{2}]_{2}$ (show 3)
Indices of inseparability: $[3,0]$
Associated inertia: $[1]$
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Label Polynomial $/ \Q_p$ Galois group $/ \Q_p$ Galois degree $/ \Q_p$ $\#\Aut(K/\Q_p)$ Artin slope content $/ \Q_p$ Swan slope content $/ \Q_p$ Hidden Artin slopes $/ \Q_p$ Hidden Swan slopes $/ \Q_p$ Ind. of Insep. $/ \Q_p$ Assoc. Inertia $/ \Q_p$ Resid. Poly Jump Set
3.2.3.10a1.1 $( x^{2} + 2 x + 2 )^{3} + 3$ $D_{6}$ (as 6T3) $12$ $2$ $[\frac{5}{2}]_{2}^{2}$ $[\frac{3}{2}]_{2}^{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[3, 0]$ $[1]$ $z + (2 t + 2)$ undefined
3.2.3.10a1.2 $( x^{2} + 2 x + 2 )^{3} + 9 x ( x^{2} + 2 x + 2 ) + 3$ $S_3^2$ (as 6T9) $36$ $1$ $[\frac{3}{2}, \frac{5}{2}]_{2}^{2}$ $[\frac{1}{2},\frac{3}{2}]_{2}^{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[3, 0]$ $[1]$ $z + (2 t + 2)$ undefined
3.2.3.10a1.3 $( x^{2} + 2 x + 2 )^{3} + 9 ( x^{2} + 2 x + 2 ) + 3$ $D_{6}$ (as 6T3) $12$ $2$ $[\frac{5}{2}]_{2}^{2}$ $[\frac{3}{2}]_{2}^{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[3, 0]$ $[1]$ $z + (2 t + 2)$ undefined
3.2.3.10a1.4 $( x^{2} + 2 x + 2 )^{3} + \left(9 x + 9\right) ( x^{2} + 2 x + 2 ) + 3$ $S_3^2$ (as 6T9) $36$ $1$ $[\frac{3}{2}, \frac{5}{2}]_{2}^{2}$ $[\frac{1}{2},\frac{3}{2}]_{2}^{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[3, 0]$ $[1]$ $z + (2 t + 2)$ undefined
3.2.3.10a1.5 $( x^{2} + 2 x + 2 )^{3} + 18 ( x^{2} + 2 x + 2 ) + 3$ $D_{6}$ (as 6T3) $12$ $2$ $[\frac{5}{2}]_{2}^{2}$ $[\frac{3}{2}]_{2}^{2}$ $[\ ]_{2}$ $[\ ]_{2}$ $[3, 0]$ $[1]$ $z + (2 t + 2)$ undefined
3.2.3.10a1.6 $( x^{2} + 2 x + 2 )^{3} + \left(9 x + 18\right) ( x^{2} + 2 x + 2 ) + 3$ $S_3^2$ (as 6T9) $36$ $1$ $[\frac{3}{2}, \frac{5}{2}]_{2}^{2}$ $[\frac{1}{2},\frac{3}{2}]_{2}^{2}$ $[\frac{3}{2}]_{2}$ $[\frac{1}{2}]_{2}$ $[3, 0]$ $[1]$ $z + (2 t + 2)$ undefined
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