$p$-adic family 2.1.6.11a1.15-2.1.0a

• Label: 2.1.6.11a1.15-2.1.0a
• Base: 2.1.6.11a1.15
• Degree: \(2\)
• e: \(1\)
• f: \(2\)
• c: \(0\)

Invariants

Residue field characteristic:	$2$
Degree:	$2$
Base field:	2.1.6.11a1.15
Ramification index $e$:	$1$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$0$
Absolute Artin slopes:	$[3]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$1$ (complete)
Ambiguity:	$2$
Mass:	$1$
Absolute Mass:	$1/4$

Varying

These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.

Galois group:	$C_2\times S_4$
Hidden Artin slopes:	$[\frac{8}{3},\frac{8}{3}]$
Indices of inseparability:	$[6,0]$
Associated inertia:	$[1,1]$
Jump Set:	$[3,9]$

Fields

Showing all 1

  Download displayed columns for results to

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
2.2.6.22a1.32	$( x^{2} + x + 1 )^{6} + 4 ( x^{2} + x + 1 )^{5} + 4 ( x^{2} + x + 1 )^{3} + 4 ( x^{2} + x + 1 ) + 2$	$C_2\times S_4$ (as 12T21)	$48$	$4$	$[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$	$[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$	$[\frac{8}{3},\frac{8}{3}]$	$[\frac{5}{3},\frac{5}{3}]$	$[6, 0]$	$[1, 1]$	$z^4 + z^2 + 1,z + 1$	$[3, 9]$

  Download displayed columns for results to