$p$-adic family 3.1.18.18a

• Label: 3.1.18.18a
• Base: 3.1.1.0a1.1
• Degree: \(18\)
• e: \(18\)
• f: \(1\)
• c: \(18\)

Defining polynomial

$x^{18} + 3 a_{1} x + 3 d_{0}$

Invariants

Residue field characteristic:	$3$
Degree:	$18$
Base field:	$\Q_{3}$
Ramification index $e$:	$18$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$18$
Artin slopes:	$[\frac{17}{16},\frac{17}{16}]$
Swan slopes:	$[\frac{1}{16},\frac{1}{16}]$
Means:	$\langle\frac{1}{24},\frac{1}{18}\rangle$
Rams:	$(\frac{1}{8},\frac{1}{8})$
Field count:	$2$ (complete)
Ambiguity:	$2$
Mass:	$2$
Absolute Mass:	$2$

Diagrams

Varying

Indices of inseparability:	$[1,1,0]$
Associated inertia:	$[1,1]$
Jump Set:	undefined (show 1), $[1,3,10]$ (show 1)

Fields

Showing all 2

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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
3.1.18.18a1.1		$x^{18} + 3 x + 3$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[1, 1, 0]$	$[1, 1]$	$z^9 + 2,2 z + 2$	$[1, 3, 10]$
3.1.18.18a1.2		$x^{18} + 3 x + 6$	not computed	$ $not computed$ $	$1$	not computed	not computed	not computed	not computed	$[1, 1, 0]$	$[1, 1]$	$z^9 + 2,2 z + 1$	undefined

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