$p$-adic family 2.1.2.2a1.1-3.2.6a

• Label: 2.1.2.2a1.1-3.2.6a
• Base: 2.1.2.2a1.1
• Degree: \(6\)
• e: \(2\)
• f: \(3\)
• c: \(6\)

Defining polynomial over unramified subextension

$x^{2} + a_{1} \pi x + c_{2} \pi^{2} + \pi$

Invariants

Residue field characteristic:	$2$
Degree:	$6$
Base field:	$\Q_{2}(\sqrt{-1})$
Ramification index $e$:	$2$
Residue field degree $f$:	$3$
Discriminant exponent $c$:	$6$
Absolute Artin slopes:	$[2,2]$
Swan slopes:	$[1]$
Means:	$\langle\frac{1}{2}\rangle$
Rams:	$(1)$
Field count:	$4$ (complete)
Ambiguity:	$6$
Mass:	$7$
Absolute Mass:	$7/6$

Diagrams

Varying

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

Galois group:	$A_4 \times C_2$ (show 1), $D_4 \times C_3$ (show 1), $C_2^2 \times A_4$ (show 1), $D_4\times A_4$ (show 1)
Hidden Artin slopes:	$[2,2]^{2}$ (show 1), $[2]$ (show 1), $[2]^{2}$ (show 1), $[\ ]^{2}$ (show 1)
Indices of inseparability:	$[3,2,0]$ (show 3), $[3,3,0]$ (show 1)
Associated inertia:	$[1]$ (show 2), $[2]$ (show 2)
Jump Set:	$[1,3,6]$

Fields

Showing all 1

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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
2.3.4.18a1.1	$( x^{3} + x + 1 )^{4} + 2 ( x^{3} + x + 1 )^{3} + 2$	$D_4 \times C_3$ (as 12T14)	$24$	$6$	$[2, 2]^{6}$	$[1,1]^{6}$	$[\ ]^{2}$	$[\ ]^{2}$	$[3, 3, 0]$	$[2]$	$z^3 + (t^2 + t)$	$[1, 3, 6]$

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