$p$-adic family 13.1.8.7a

• Label: 13.1.8.7a
• Base: 13.1.1.0a1.1
• Degree: \(8\)
• e: \(8\)
• f: \(1\)
• c: \(7\)

Defining polynomial

$x^{8} + 13d_{0}$

Invariants

Residue field characteristic:	$13$
Degree:	$8$
Base field:	$\Q_{13}$
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

Galois groups and Hidden Artin slopes

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

Fields

Showing all 4

  Download displayed columns for results to

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
13.1.8.7a1.1	4	$x^{8} + 13$	$C_8:C_2$ (as 8T7)	$16$	$4$	$[\ ]_{8}^{2}$	$[\ ]_{8}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^7 + 8 z^6 + 2 z^5 + 4 z^4 + 5 z^3 + 4 z^2 + 2 z + 8$	undefined
13.1.8.7a1.2	4	$x^{8} + 26$	$C_8:C_2$ (as 8T7)	$16$	$4$	$[\ ]_{8}^{2}$	$[\ ]_{8}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^7 + 8 z^6 + 2 z^5 + 4 z^4 + 5 z^3 + 4 z^2 + 2 z + 8$	undefined
13.1.8.7a1.3	4	$x^{8} + 52$	$C_8:C_2$ (as 8T7)	$16$	$4$	$[\ ]_{8}^{2}$	$[\ ]_{8}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^7 + 8 z^6 + 2 z^5 + 4 z^4 + 5 z^3 + 4 z^2 + 2 z + 8$	undefined
13.1.8.7a1.4	4	$x^{8} + 104$	$C_8:C_2$ (as 8T7)	$16$	$4$	$[\ ]_{8}^{2}$	$[\ ]_{8}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^7 + 8 z^6 + 2 z^5 + 4 z^4 + 5 z^3 + 4 z^2 + 2 z + 8$	undefined

  Download displayed columns for results to