$p$-adic family 19.1.3.2a

• Label: 19.1.3.2a
• Base: 19.1.1.0a1.1
• Degree: \(3\)
• e: \(3\)
• f: \(1\)
• c: \(2\)

Defining polynomial

$x^{3} + 19d_{0}$

Invariants

Residue field characteristic:	$19$
Degree:	$3$
Base field:	$\Q_{19}$
Ramification index $e$:	$3$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$2$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$3$ (complete)
Ambiguity:	$3$
Mass:	$1$
Absolute Mass:	$1$

Varying

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

Galois groups and Hidden Artin slopes

		Galois groups
		$C_3$ (as 3T1)
hidden slopes	$[\ ]$	3

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
19.1.3.2a1.1	3	$x^{3} + 19$	$C_3$ (as 3T1)	$3$	$3$	$[\ ]_{3}$	$[\ ]_{3}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^2 + 3 z + 3$	undefined
19.1.3.2a1.2	3	$x^{3} + 38$	$C_3$ (as 3T1)	$3$	$3$	$[\ ]_{3}$	$[\ ]_{3}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^2 + 3 z + 3$	undefined
19.1.3.2a1.3	3	$x^{3} + 76$	$C_3$ (as 3T1)	$3$	$3$	$[\ ]_{3}$	$[\ ]_{3}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^2 + 3 z + 3$	undefined

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