$p$-adic family 2.2.3.4a

• Label: 2.2.3.4a
• Base: 2.1.1.0a1.1
• Degree: \(6\)
• e: \(3\)
• f: \(2\)
• c: \(4\)

Defining polynomial over unramified subextension

$x^{3} + 2d_{0}$

Invariants

Residue field characteristic:	$2$
Degree:	$6$
Base field:	$\Q_{2}$
Ramification index $e$:	$3$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$4$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$2$ (complete)
Ambiguity:	$6$
Mass:	$1$
Absolute Mass:	$1/2$

Varying

Indices of inseparability:	$[0]$
Associated inertia:	$[1]$
Jump Set:	$[3]$

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 6
		$S_3$ (as 6T2)
hidden slopes	$[\ ]$	1

		Galois groups of order 18
		$C_3\times S_3$ (as 6T5)
hidden slopes	$[\ ]^{3}$	1

Fields

Showing all 2

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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
2.2.3.4a1.1	1	$( x^{2} + x + 1 )^{3} + 2 x$	$S_3\times C_3$ (as 6T5)	$18$	$3$	$[\ ]_{3}^{6}$	$[\ ]_{3}^{6}$	$[\ ]^{3}$	$[\ ]^{3}$	$[0]$	$[1]$	$z^2 + z + 1$	$[3]$
2.2.3.4a1.2	1	$( x^{2} + x + 1 )^{3} + 2$	$S_3$ (as 6T2)	$6$	$6$	$[\ ]_{3}^{2}$	$[\ ]_{3}^{2}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^2 + z + 1$	$[3]$

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