$p$-adic family 5.2.9.16a

• Label: 5.2.9.16a
• Base: 5.1.1.0a1.1
• Degree: \(18\)
• e: \(9\)
• f: \(2\)
• c: \(16\)

Defining polynomial over unramified subextension

$x^{9} + 5d_{0}$

Invariants

Residue field characteristic:	$5$
Degree:	$18$
Base field:	$\Q_{5}$
Ramification index $e$:	$9$
Residue field degree $f$:	$2$
Discriminant exponent $c$:	$16$
Artin slopes:	$[\ ]$
Swan slopes:	$[\ ]$
Means:	$\langle\ \rangle$
Rams:	$(\ )$
Field count:	$2$ (complete)
Ambiguity:	$6$
Mass:	$1$
Absolute Mass:	$1/2$

Varying

Indices of inseparability:	$[0]$
Associated inertia:	$[3]$
Jump Set:	undefined

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 54
		$C_9:C_6$ (as 18T18)
hidden slopes	$[\ ]^{3}$	1

		Galois groups of order 162
		$C_9:C_{18}$ (as 18T80)
hidden slopes	$[\ ]^{9}$	1

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
5.2.9.16a1.1	1	$( x^{2} + 4 x + 2 )^{9} + 5 x$	$C_9:C_{18}$ (as 18T80)	$162$	$3$	$[\ ]_{9}^{18}$	$[\ ]_{9}^{18}$	$[\ ]^{9}$	$[\ ]^{9}$	$[0]$	$[3]$	$z^8 + 4 z^7 + z^6 + 4 z^5 + z^4 + z^3 + 4 z^2 + z + 4$	undefined
5.2.9.16a1.2	1	$( x^{2} + 4 x + 2 )^{9} + 5$	$C_9:C_6$ (as 18T18)	$54$	$6$	$[\ ]_{9}^{6}$	$[\ ]_{9}^{6}$	$[\ ]^{3}$	$[\ ]^{3}$	$[0]$	$[3]$	$z^8 + 4 z^7 + z^6 + 4 z^5 + z^4 + z^3 + 4 z^2 + z + 4$	undefined

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