$p$-adic family 5.1.8.7a

• Label: 5.1.8.7a
• Base: 5.1.1.0a1.1
• Degree: \(8\)
• e: \(8\)
• f: \(1\)
• c: \(7\)

Defining polynomial

$x^{8} + 5d_{0}$

Invariants

Residue field characteristic:	$5$
Degree:	$8$
Base field:	$\Q_{5}$
Ramification index $e$:	$8$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$7$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$4$ (complete)
Ambiguity:	$4$
Mass:	$1$
Absolute Mass:	$1$

Varying

Indices of inseparability:	$[0]$
Associated inertia:	$[2]$
Jump Set:	undefined (show 3), $[2]$ (show 1)

Galois groups and Hidden Artin slopes

		Galois groups
		$\OD_{16}$ (as 8T7)
hidden slopes	$[\ ]^{2}$	4

Fields

Showing all 4

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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
5.1.8.7a1.1	4	$x^{8} + 5$	$C_8:C_2$ (as 8T7)	$16$	$4$	$[\ ]_{8}^{2}$	$[\ ]_{8}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^7 + 3 z^6 + 3 z^5 + z^4 + z^2 + 3 z + 3$	$[2]$
5.1.8.7a1.2	4	$x^{8} + 10$	$C_8:C_2$ (as 8T7)	$16$	$4$	$[\ ]_{8}^{2}$	$[\ ]_{8}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^7 + 3 z^6 + 3 z^5 + z^4 + z^2 + 3 z + 3$	undefined
5.1.8.7a1.3	4	$x^{8} + 15$	$C_8:C_2$ (as 8T7)	$16$	$4$	$[\ ]_{8}^{2}$	$[\ ]_{8}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^7 + 3 z^6 + 3 z^5 + z^4 + z^2 + 3 z + 3$	undefined
5.1.8.7a1.4	4	$x^{8} + 20$	$C_8:C_2$ (as 8T7)	$16$	$4$	$[\ ]_{8}^{2}$	$[\ ]_{8}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^7 + 3 z^6 + 3 z^5 + z^4 + z^2 + 3 z + 3$	undefined

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