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Properties

Label 3.2.1.0a1.1-1.3.4a
Base 3.2.1.0a1.1
Degree \(3\)
e \(3\)
f \(1\)
c \(4\)

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Defining polynomial

$x^{3} + 3 a_{2} x^{2} + 9 c_{3} + 3$

Invariants

Residue field characteristic: $3$
Degree: $3$
Base field: $\Q_{3}(\sqrt{2})$
Ramification index $e$: $3$
Residue field degree $f$: $1$
Discriminant exponent $c$: $4$
Absolute Artin slopes: $[2]$
Swan slopes: $[1]$
Means: $\langle\frac{2}{3}\rangle$
Rams: $(1)$
Field count: $10$ (complete)
Ambiguity: $3$
Mass: $8$
Absolute Mass: $4$

Diagrams

Varying

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

Galois group: $C_6$ (show 3), $S_3$ (show 1), $S_3\times C_3$ (show 4), $C_3^2:C_4$ (show 2)
Hidden Artin slopes: $[2]^{2}$ (show 2), $[\ ]$ (show 4), $[\ ]^{3}$ (show 1), $[2]$ (show 3)
Indices of inseparability: $[2,0]$
Associated inertia: $[1]$ (show 8), $[2]$ (show 2)
Jump Set: undefined

Fields

Galois group Galois Artin Indices
Assoc. Inertia Jump Set
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Showing all 8

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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
3.2.3.8a1.1 $( x^{2} + 2 x + 2 )^{3} + 3 ( x^{2} + 2 x + 2 )^{2} + 3$ $S_3$ (as 6T2) $6$ $6$ $[2]^{2}$ $[1]^{2}$ $[\ ]$ $[\ ]$ $[2, 0]$ $[1]$ $z^2 + 2$ undefined
3.2.3.8a1.2 $( x^{2} + 2 x + 2 )^{3} + 3 ( x^{2} + 2 x + 2 )^{2} + 9 x + 3$ $S_3\times C_3$ (as 6T5) $18$ $3$ $[2]^{6}$ $[1]^{6}$ $[\ ]^{3}$ $[\ ]^{3}$ $[2, 0]$ $[1]$ $z^2 + 2$ undefined
3.2.3.8a2.1 $( x^{2} + 2 x + 2 )^{3} + 6 ( x^{2} + 2 x + 2 )^{2} + 3$ $C_6$ (as 6T1) $6$ $6$ $[2]^{2}$ $[1]^{2}$ $[\ ]$ $[\ ]$ $[2, 0]$ $[1]$ $z^2 + 1$ undefined
3.2.3.8a2.2 $( x^{2} + 2 x + 2 )^{3} + 6 ( x^{2} + 2 x + 2 )^{2} + 12$ $C_6$ (as 6T1) $6$ $6$ $[2]^{2}$ $[1]^{2}$ $[\ ]$ $[\ ]$ $[2, 0]$ $[1]$ $z^2 + 1$ undefined
3.2.3.8a2.3 $( x^{2} + 2 x + 2 )^{3} + 6 ( x^{2} + 2 x + 2 )^{2} + 21$ $C_6$ (as 6T1) $6$ $6$ $[2]^{2}$ $[1]^{2}$ $[\ ]$ $[\ ]$ $[2, 0]$ $[1]$ $z^2 + 1$ undefined
3.2.3.8a4.1 $( x^{2} + 2 x + 2 )^{3} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{2} + 3$ $S_3\times C_3$ (as 6T5) $18$ $3$ $[2, 2]^{2}$ $[1,1]^{2}$ $[2]$ $[1]$ $[2, 0]$ $[1]$ $z^2 + (2 t + 2)$ undefined
3.2.3.8a4.2 $( x^{2} + 2 x + 2 )^{3} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{2} + 12$ $S_3\times C_3$ (as 6T5) $18$ $3$ $[2, 2]^{2}$ $[1,1]^{2}$ $[2]$ $[1]$ $[2, 0]$ $[1]$ $z^2 + (2 t + 2)$ undefined
3.2.3.8a4.3 $( x^{2} + 2 x + 2 )^{3} + \left(3 x + 3\right) ( x^{2} + 2 x + 2 )^{2} + 21$ $S_3\times C_3$ (as 6T5) $18$ $3$ $[2, 2]^{2}$ $[1,1]^{2}$ $[2]$ $[1]$ $[2, 0]$ $[1]$ $z^2 + (2 t + 2)$ undefined
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