$p$-adic family 2.1.2.3a1.1-1.2.5a

• Label: 2.1.2.3a1.1-1.2.5a
• Base: 2.1.2.3a1.1
• Degree: \(2\)
• e: \(2\)
• f: \(1\)
• c: \(5\)

Defining polynomial

$x^{2} + \left(b_{7} \pi^{4} + b_{5} \pi^{3}\right) x + c_{8} \pi^{5} + \pi$

Invariants

Residue field characteristic:	$2$
Degree:	$2$
Base field:	$\Q_{2}(\sqrt{-2})$
Ramification index $e$:	$2$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$5$
Absolute Artin slopes:	$[3,4]$
Swan slopes:	$[4]$
Means:	$\langle2\rangle$
Rams:	$(4)$
Field count:	$4$ (complete)
Ambiguity:	$2$
Mass:	$4$
Absolute Mass:	$2$

Diagrams

Varying

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

Galois group:	$D_{4}$
Hidden Artin slopes:	$[2]$ (show 2), $[\ ]^{2}$ (show 2)
Indices of inseparability:	$[8,4,0]$
Associated inertia:	$[1,1]$
Jump Set:	$[1,3,7]$

Fields

Showing all 4

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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
2.1.4.11a1.1	$x^{4} + 8 x^{3} + 2$	$D_{4}$ (as 4T3)	$8$	$2$	$[2, 3, 4]$	$[1,2,3]$	$[2]$	$[1]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.2	$x^{4} + 8 x^{3} + 18$	$D_{4}$ (as 4T3)	$8$	$2$	$[2, 3, 4]$	$[1,2,3]$	$[2]$	$[1]$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.3	$x^{4} + 8 x + 18$	$D_{4}$ (as 4T3)	$8$	$2$	$[3, 4]^{2}$	$[2,3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$
2.1.4.11a1.4	$x^{4} + 8 x^{3} + 8 x + 18$	$D_{4}$ (as 4T3)	$8$	$2$	$[3, 4]^{2}$	$[2,3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[8, 4, 0]$	$[1, 1]$	$z^2 + 1,z + 1$	$[1, 3, 7]$

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