$p$-adic family 19.6.1.0a1.1-1.3.2a

• Label: 19.6.1.0a1.1-1.3.2a
• Base: 19.6.1.0a1.1
• Degree: \(3\)
• e: \(3\)
• f: \(1\)
• c: \(2\)

Defining polynomial

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

Invariants

Residue field characteristic:	$19$
Degree:	$3$
Base field:	19.6.1.0a1.1
Ramification index $e$:	$3$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$2$
Absolute Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$3$ (complete)
Ambiguity:	$3$
Mass:	$1$
Absolute Mass:	$1/6$

Varying

These invariants are all associated to absolute extensions of $\Q_{ 19 }$ within this relative family, not the relative extension.

Galois group:	$C_{18}$ (show 2), $C_6 \times C_3$ (show 1)
Hidden Artin slopes:	$[\ ]$
Indices of inseparability:	$[0]$
Associated inertia:	$[1]$
Jump Set:	undefined

Fields

Showing all 3

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Label	Polynomial $/ \Q_p$	Galois group $/ \Q_p$	Galois degree $/ \Q_p$	$\#\Aut(K/\Q_p)$	Artin slope content $/ \Q_p$	Swan slope content $/ \Q_p$	Hidden Artin slopes $/ \Q_p$	Hidden Swan slopes $/ \Q_p$	Ind. of Insep. $/ \Q_p$	Assoc. Inertia $/ \Q_p$	Resid. Poly	Jump Set
19.6.3.12a1.1	$( x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2 )^{3} + 19 x^{2}$	$C_{18}$ (as 18T1)	$18$	$18$	$[\ ]_{3}^{6}$	$[\ ]_{3}^{6}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^2 + 3 z + 3$	undefined
19.6.3.12a1.2	$( x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2 )^{3} + 19 x$	$C_{18}$ (as 18T1)	$18$	$18$	$[\ ]_{3}^{6}$	$[\ ]_{3}^{6}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^2 + 3 z + 3$	undefined
19.6.3.12a1.3	$( x^{6} + 17 x^{3} + 17 x^{2} + 6 x + 2 )^{3} + 19$	$C_6 \times C_3$ (as 18T2)	$18$	$18$	$[\ ]_{3}^{6}$	$[\ ]_{3}^{6}$	$[\ ]$	$[\ ]$	$[0]$	$[1]$	$z^2 + 3 z + 3$	undefined

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