$p$-adic family 3.1.9.9a1.1-1.2.1a

• Label: 3.1.9.9a1.1-1.2.1a
• Base: 3.1.9.9a1.1
• Degree: \(2\)
• e: \(2\)
• f: \(1\)
• c: \(1\)

Defining polynomial

$x^{2} + d_{0} \pi$

Invariants

Residue field characteristic:	$3$
Degree:	$2$
Base field:	3.1.9.9a1.1
Ramification index $e$:	$2$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$1$
Absolute Artin slopes:	$[\frac{9}{8},\frac{9}{8}]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$2$ (complete)
Ambiguity:	$2$
Mass:	$1$
Absolute Mass:	$1$

Varying

These invariants are all associated to absolute extensions of $\Q_{ 3 }$ within this relative family, not the relative extension.

Indices of inseparability:	$[2,2,0]$
Associated inertia:	$[1,1]$ (show 1), $[1,2]$ (show 1)
Jump Set:	undefined (show 1), $[1,3,11]$ (show 1)

Fields

Showing all 2

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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
3.1.18.19a1.1	$x^{18} + 3 x^{2} + 3$	$F_9:C_2$ (as 18T73)	$144$	$2$	not computed	not computed	not computed	not computed	$[2, 2, 0]$	$[1, 1]$	$z^9 + 2,2 z^2 + 1$	$[1, 3, 11]$
3.1.18.19a2.2	$x^{18} + 3 x^{2} + 6$	$F_9:C_2$ (as 18T68)	$144$	$2$	not computed	not computed	not computed	not computed	$[2, 2, 0]$	$[1, 2]$	$z^9 + 2,2 z^2 + 2$	undefined

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