$p$-adic family 5.1.15.16a

• Label: 5.1.15.16a
• Base: 5.1.1.0a1.1
• Degree: \(15\)
• e: \(15\)
• f: \(1\)
• c: \(16\)

Defining polynomial

$x^{15} + 5a_{2} x^{2} + 5$

Invariants

Residue field characteristic:	$5$
Degree:	$15$
Base field:	$\Q_{5}$
Ramification index $e$:	$15$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$16$
Artin slopes:	$[\frac{7}{6}]$
Swan slopes:	$[\frac{1}{6}]$
Means:	$\langle\frac{2}{15}\rangle$
Rams:	$(\frac{1}{2})$
Field count:	$4$ (complete)
Ambiguity:	$1$
Mass:	$4$
Absolute Mass:	$4$

Diagrams

Varying

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

Galois groups and Hidden Artin slopes

		Galois groups
		$C_5^2:C_3:C_4$ (as 15T17)	$C_5^2:D_6$ (as 15T18)	Total
hidden slopes	$[\frac{7}{6}]^{2}_{2}$	2	2	4

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
5.1.15.16a2.1	2	$x^{15} + 5 x^{2} + 5$	$((C_5^2 : C_3):C_2):C_2$ (as 15T18)	$300$	$1$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]^{2}_{2}$	$[\frac{1}{6}]^{2}_{2}$	$[2, 0]$	$[2, 1]$	$z^{10} + 3 z^5 + 3,3 z^2 + 3$	undefined
5.1.15.16a2.2	2	$x^{15} + 20 x^{2} + 5$	$((C_5^2 : C_3):C_2):C_2$ (as 15T18)	$300$	$1$	$[\frac{7}{6}, \frac{7}{6}]_{6}^{2}$	$[\frac{1}{6},\frac{1}{6}]_{6}^{2}$	$[\frac{7}{6}]^{2}_{2}$	$[\frac{1}{6}]^{2}_{2}$	$[2, 0]$	$[2, 1]$	$z^{10} + 3 z^5 + 3,3 z^2 + 2$	undefined

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