$p$-adic family 7.1.16.15a

• Label: 7.1.16.15a
• Base: 7.1.1.0a1.1
• Degree: \(16\)
• e: \(16\)
• f: \(1\)
• c: \(15\)

Defining polynomial

$x^{16} + 7d_{0}$

Invariants

Residue field characteristic:	$7$
Degree:	$16$
Base field:	$\Q_{7}$
Ramification index $e$:	$16$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$15$
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
		$\SD_{32}$ (as 16T55)
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
7.1.16.15a1.1	2	$x^{16} + 7$	$\SD_{32}$ (as 16T55)	$32$	$2$	$[\ ]_{16}^{2}$	$[\ ]_{16}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^{15} + 2 z^{14} + z^{13} + 2 z^8 + 4 z^7 + 2 z^6 + z + 2$	undefined
7.1.16.15a1.2	2	$x^{16} + 21$	$\SD_{32}$ (as 16T55)	$32$	$2$	$[\ ]_{16}^{2}$	$[\ ]_{16}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[0]$	$[2]$	$z^{15} + 2 z^{14} + z^{13} + 2 z^8 + 4 z^7 + 2 z^6 + z + 2$	undefined

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