$p$-adic family 3.1.18.19a

• Label: 3.1.18.19a
• Base: 3.1.1.0a1.1
• Degree: \(18\)
• e: \(18\)
• f: \(1\)
• c: \(19\)

Defining polynomial

$x^{18} + 3 a_{2} x^{2} + 3 d_{0}$

Invariants

Residue field characteristic:	$3$
Degree:	$18$
Base field:	$\Q_{3}$
Ramification index $e$:	$18$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$19$
Artin slopes:	$[\frac{9}{8},\frac{9}{8}]$
Swan slopes:	$[\frac{1}{8},\frac{1}{8}]$
Means:	$\langle\frac{1}{12},\frac{1}{9}\rangle$
Rams:	$(\frac{1}{4},\frac{1}{4})$
Field count:	$4$ (complete)
Ambiguity:	$2$
Mass:	$2$
Absolute Mass:	$2$

Diagrams

Varying

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

Fields

Showing all 4

  Download displayed columns for results to

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

  Download displayed columns for results to