$p$-adic family 17.3.4.9a

• Label: 17.3.4.9a
• Base: 17.1.1.0a1.1
• Degree: \(12\)
• e: \(4\)
• f: \(3\)
• c: \(9\)

Defining polynomial over unramified subextension

$x^{4} + 17d_{0}$

Invariants

Residue field characteristic:	$17$
Degree:	$12$
Base field:	$\Q_{17}$
Ramification index $e$:	$4$
Residue field degree $f$:	$3$
Discriminant exponent $c$:	$9$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$4$ (complete)
Ambiguity:	$12$
Mass:	$1$
Absolute Mass:	$1/3$

Varying

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

Galois groups and Hidden Artin slopes

		Galois groups
		$C_{12}$ (as 12T1)
hidden slopes	$[\ ]$	4

Fields

Showing all 4

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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
17.3.4.9a1.1	4	$( x^{3} + x + 14 )^{4} + 17 x^{2}$	$C_{12}$ (as 12T1)	$12$	$12$	$[\ ]_{4}^{3}$	$[\ ]_{4}^{3}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^3 + 4 z^2 + 6 z + 4$	undefined
17.3.4.9a1.2	4	$( x^{3} + x + 14 )^{4} + 17 x$	$C_{12}$ (as 12T1)	$12$	$12$	$[\ ]_{4}^{3}$	$[\ ]_{4}^{3}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^3 + 4 z^2 + 6 z + 4$	undefined
17.3.4.9a1.3	4	$( x^{3} + x + 14 )^{4} + 17$	$C_{12}$ (as 12T1)	$12$	$12$	$[\ ]_{4}^{3}$	$[\ ]_{4}^{3}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^3 + 4 z^2 + 6 z + 4$	undefined
17.3.4.9a1.4	4	$( x^{3} + x + 14 )^{4} + 272 x + 51$	$C_{12}$ (as 12T1)	$12$	$12$	$[\ ]_{4}^{3}$	$[\ ]_{4}^{3}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^3 + 4 z^2 + 6 z + 4$	undefined

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