$p$-adic family 3.4.4.12a

• Label: 3.4.4.12a
• Base: 3.1.1.0a1.1
• Degree: \(16\)
• e: \(4\)
• f: \(4\)
• c: \(12\)

Defining polynomial over unramified subextension

$x^{4} + 3d_{0}$

Invariants

Residue field characteristic:	$3$
Degree:	$16$
Base field:	$\Q_{3}$
Ramification index $e$:	$4$
Residue field degree $f$:	$4$
Discriminant exponent $c$:	$12$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$3$ (complete)
Ambiguity:	$16$
Mass:	$1$
Absolute Mass:	$1/4$

Varying

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

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 16
		$\OD_{16}$ (as 16T6)	$C_4:C_4$ (as 16T8)	Total
hidden slopes	$[\ ]$	1	1	2

		Galois groups of order 32
		$\OD_{32}$ (as 16T22)
hidden slopes	$[\ ]^{2}$	1

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
3.4.4.12a1.1	1	$( x^{4} + 2 x^{3} + 2 )^{4} + 3 x^{2}$	$C_8: C_2$ (as 16T6)	$16$	$16$	$[\ ]_{4}^{4}$	$[\ ]_{4}^{4}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^3 + z^2 + 1$	$[2]$
3.4.4.12a1.2	1	$( x^{4} + 2 x^{3} + 2 )^{4} + 3 x$	$C_{16} : C_2$ (as 16T22)	$32$	$8$	$[\ ]_{4}^{8}$	$[\ ]_{4}^{8}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[1]$	$z^3 + z^2 + 1$	undefined
3.4.4.12a1.3	1	$( x^{4} + 2 x^{3} + 2 )^{4} + 3$	$C_4:C_4$ (as 16T8)	$16$	$16$	$[\ ]_{4}^{4}$	$[\ ]_{4}^{4}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^3 + z^2 + 1$	$[2]$

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