$p$-adic family 3.1.3.4a

• Label: 3.1.3.4a
• Base: 3.1.1.0a1.1
• Degree: \(3\)
• e: \(3\)
• f: \(1\)
• c: \(4\)

Defining polynomial

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

Invariants

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

Diagrams

Varying

Indices of inseparability:	$[2,0]$
Associated inertia:	$[1]$ (show 3), $[2]$ (show 1)
Jump Set:	undefined

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 3

		Galois groups of order 6

Fields

Showing all 3

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Label	Packet size	Polynomial	Galois group	Galois degree	$\#\Aut(K/\Q_p)$	Artin slope content	Swan slope content	Hidden Artin slopes	Hidden Swan slopes	Ind. of Insep.	Assoc. Inertia	Resid. Poly	Jump Set
3.1.3.4a2.1	3	$x^{3} + 6 x^{2} + 3$	$C_3$ (as 3T1)	$3$	$3$	$[2]$	$[1]$	$[\ ]$	$[\ ]$	$[2, 0]$	$[1]$	$z^2 + 2$	undefined
3.1.3.4a2.2	3	$x^{3} + 6 x^{2} + 12$	$C_3$ (as 3T1)	$3$	$3$	$[2]$	$[1]$	$[\ ]$	$[\ ]$	$[2, 0]$	$[1]$	$z^2 + 2$	undefined
3.1.3.4a2.3	3	$x^{3} + 6 x^{2} + 21$	$C_3$ (as 3T1)	$3$	$3$	$[2]$	$[1]$	$[\ ]$	$[\ ]$	$[2, 0]$	$[1]$	$z^2 + 2$	undefined

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