$p$-adic family 3.1.6.7a1.2-1.3.3a

• Label: 3.1.6.7a1.2-1.3.3a
• Base: 3.1.6.7a1.2
• Degree: \(3\)
• e: \(3\)
• f: \(1\)
• c: \(3\)

Defining polynomial

$x^{3} + a_{1} \pi x + \pi$

Invariants

Residue field characteristic:	$3$
Degree:	$3$
Base field:	3.1.6.7a1.2
Ramification index $e$:	$3$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$3$
Absolute Artin slopes:	$[\frac{5}{4},\frac{3}{2}]$
Swan slopes:	$[\frac{1}{2}]$
Means:	$\langle\frac{1}{3}\rangle$
Rams:	$(\frac{1}{2})$
Field count:	$2$ (complete)
Ambiguity:	$1$
Mass:	$2$
Absolute Mass:	$2/3$

Diagrams

Varying

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

Indices of inseparability:	$[7,3,0]$
Associated inertia:	$[1,1,1]$
Jump Set:	$[1,4,16]$

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
3.1.18.24b2.7	$x^{18} + 6 x^{8} + 6 x^{7} + 3 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[7, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 1$	$[1, 4, 16]$
3.1.18.24b2.9	$x^{18} + 6 x^{9} + 6 x^{8} + 6 x^{7} + 3 x^{3} + 3$	not computed	$ $not computed$ $	$3$	not computed	not computed	not computed	not computed	$[7, 3, 0]$	$[1, 1, 1]$	$z^9 + 2,2 z^3 + 2,2 z^2 + 1$	$[1, 4, 16]$

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