$p$-adic family 5.1.12.11a

• Label: 5.1.12.11a
• Base: 5.1.1.0a1.1
• Degree: \(12\)
• e: \(12\)
• f: \(1\)
• c: \(11\)

Defining polynomial

$x^{12} + 5d_{0}$

Invariants

Residue field characteristic:	$5$
Degree:	$12$
Base field:	$\Q_{5}$
Ramification index $e$:	$12$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$11$
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), $[3]$ (show 1)

Galois groups and Hidden Artin slopes

		Galois groups
		$C_4\times S_3$ (as 12T11)
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
5.1.12.11a1.1	4	$x^{12} + 5$	$S_3 \times C_4$ (as 12T11)	$24$	$4$	$[\ ]_{12}^{2}$	$[\ ]_{12}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^{11} + 2 z^{10} + z^9 + 2 z^6 + 4 z^5 + 2 z^4 + z + 2$	$[3]$
5.1.12.11a1.2	4	$x^{12} + 10$	$S_3 \times C_4$ (as 12T11)	$24$	$4$	$[\ ]_{12}^{2}$	$[\ ]_{12}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^{11} + 2 z^{10} + z^9 + 2 z^6 + 4 z^5 + 2 z^4 + z + 2$	undefined
5.1.12.11a1.3	4	$x^{12} + 15$	$S_3 \times C_4$ (as 12T11)	$24$	$4$	$[\ ]_{12}^{2}$	$[\ ]_{12}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^{11} + 2 z^{10} + z^9 + 2 z^6 + 4 z^5 + 2 z^4 + z + 2$	undefined
5.1.12.11a1.4	4	$x^{12} + 20$	$S_3 \times C_4$ (as 12T11)	$24$	$4$	$[\ ]_{12}^{2}$	$[\ ]_{12}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^{11} + 2 z^{10} + z^9 + 2 z^6 + 4 z^5 + 2 z^4 + z + 2$	undefined

  Download displayed columns for results to