$p$-adic family 67.1.8.7a

• Label: 67.1.8.7a
• Base: 67.1.1.0a1.1
• Degree: \(8\)
• e: \(8\)
• f: \(1\)
• c: \(7\)

Defining polynomial

$x^{8} + 67d_{0}$

Invariants

Residue field characteristic:	$67$
Degree:	$8$
Base field:	$\Q_{67}$
Ramification index $e$:	$8$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$7$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$2$ (complete)
Ambiguity:	$2$
Mass:	$1$
Absolute Mass:	$1$

Varying

Indices of inseparability:	$[0]$
Associated inertia:	$[2]$
Jump Set:	undefined

Galois groups and Hidden Artin slopes

		Galois groups
		$\SD_{16}$ (as 8T8)
hidden slopes	$[\ ]^{2}$	2

Fields

Showing all 2

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Label	Packet size	Polynomial	Galois group	Galois degree	$\#\Aut(K/\Q_p)$	Artin slope content	Swan slope content	Hidden Artin slopes	Hidden Swan slopes	Ind. of Insep.	Assoc. Inertia	Resid. Poly	Jump Set
67.1.8.7a1.1	2	$x^{8} + 67$	$QD_{16}$ (as 8T8)	$16$	$2$	$[\ ]_{8}^{2}$	$[\ ]_{8}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^7 + 8 z^6 + 28 z^5 + 56 z^4 + 3 z^3 + 56 z^2 + 28 z + 8$	undefined
67.1.8.7a1.2	2	$x^{8} + 134$	$QD_{16}$ (as 8T8)	$16$	$2$	$[\ ]_{8}^{2}$	$[\ ]_{8}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^7 + 8 z^6 + 28 z^5 + 56 z^4 + 3 z^3 + 56 z^2 + 28 z + 8$	undefined

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