$p$-adic family 2.1.2.3a1.3-1.4.8a

• Label: 2.1.2.3a1.3-1.4.8a
• Base: 2.1.2.3a1.3
• Degree: \(4\)
• e: \(4\)
• f: \(1\)
• c: \(8\)

Defining polynomial

$x^{4} + b_{6} \pi^{2} x^{2} + a_{5} \pi^{2} x + \pi$

Invariants

Residue field characteristic:	$2$
Degree:	$4$
Base field:	$\Q_{2}(\sqrt{2})$
Ramification index $e$:	$4$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$8$
Absolute Artin slopes:	$[\frac{8}{3},\frac{8}{3},3]$
Swan slopes:	$[\frac{5}{3},\frac{5}{3}]$
Means:	$\langle\frac{5}{6},\frac{5}{4}\rangle$
Rams:	$(\frac{5}{3},\frac{5}{3})$
Field count:	$2$ (complete)
Ambiguity:	$1$
Mass:	$2$
Absolute Mass:	$1$

Diagrams

Varying

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

Galois group:	$S_4\times C_2$
Hidden Artin slopes:	$[\ ]^{2}_{3}$
Indices of inseparability:	$[13,10,8,0]$
Associated inertia:	$[1,1]$
Jump Set:	$[1,3,7,15]$

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
2.1.8.20c1.3	$x^{8} + 4 x^{7} + 4 x^{5} + 4 x^{2} + 2$	$S_4\times C_2$ (as 8T24)	$48$	$2$	$[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$	$[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$	$[\ ]^{2}_{3}$	$[\ ]^{2}_{3}$	$[13, 10, 8, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15]$
2.1.8.20c1.5	$x^{8} + 4 x^{5} + 4 x^{4} + 4 x^{2} + 2$	$S_4\times C_2$ (as 8T24)	$48$	$2$	$[\frac{8}{3}, \frac{8}{3}, 3]_{3}^{2}$	$[\frac{5}{3},\frac{5}{3},2]_{3}^{2}$	$[\ ]^{2}_{3}$	$[\ ]^{2}_{3}$	$[13, 10, 8, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7, 15]$

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