$p$-adic family 7.1.14.26a

• Label: 7.1.14.26a
• Base: 7.1.1.0a1.1
• Degree: \(14\)
• e: \(14\)
• f: \(1\)
• c: \(26\)

Defining polynomial

$x^{14} + 7 a_{13} x^{13} + 49 b_{15} x + 7 d_{0}$

Invariants

Residue field characteristic:	$7$
Degree:	$14$
Base field:	$\Q_{7}$
Ramification index $e$:	$14$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$26$
Artin slopes:	$[\frac{25}{12}]$
Swan slopes:	$[\frac{13}{12}]$
Means:	$\langle\frac{13}{14}\rangle$
Rams:	$(\frac{13}{6})$
Field count:	$42$ (complete)
Ambiguity:	$2$
Mass:	$42$
Absolute Mass:	$42$

Diagrams

Varying

Indices of inseparability:	$[13,0]$
Associated inertia:	$[1,1]$
Jump Set:	undefined

Galois groups and Hidden Artin slopes

		Galois groups
		$D_7^2:C_6$ (as 14T32)
hidden slopes	$[\frac{25}{12}]^{2}_{6}$	42

Fields

Showing all 42

  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
7.1.14.26a1.1	42	$x^{14} + 7 x^{13} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 1$	undefined
7.1.14.26a1.2	42	$x^{14} + 7 x^{13} + 49 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 1$	undefined
7.1.14.26a1.3	42	$x^{14} + 7 x^{13} + 98 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 1$	undefined
7.1.14.26a1.4	42	$x^{14} + 7 x^{13} + 147 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 1$	undefined
7.1.14.26a1.5	42	$x^{14} + 14 x^{13} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 2$	undefined
7.1.14.26a1.6	42	$x^{14} + 14 x^{13} + 49 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 2$	undefined
7.1.14.26a1.7	42	$x^{14} + 14 x^{13} + 98 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 2$	undefined
7.1.14.26a1.8	42	$x^{14} + 14 x^{13} + 147 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 2$	undefined
7.1.14.26a1.9	42	$x^{14} + 21 x^{13} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 3$	undefined
7.1.14.26a1.10	42	$x^{14} + 21 x^{13} + 49 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 3$	undefined
7.1.14.26a1.11	42	$x^{14} + 21 x^{13} + 98 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 3$	undefined
7.1.14.26a1.12	42	$x^{14} + 21 x^{13} + 147 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 3$	undefined
7.1.14.26a1.13	42	$x^{14} + 28 x^{13} + 49 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 4$	undefined
7.1.14.26a1.14	42	$x^{14} + 28 x^{13} + 98 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 4$	undefined
7.1.14.26a1.15	42	$x^{14} + 28 x^{13} + 147 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 4$	undefined
7.1.14.26a1.16	42	$x^{14} + 35 x^{13} + 49 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 5$	undefined
7.1.14.26a1.17	42	$x^{14} + 35 x^{13} + 98 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 5$	undefined
7.1.14.26a1.18	42	$x^{14} + 35 x^{13} + 147 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 5$	undefined
7.1.14.26a1.19	42	$x^{14} + 42 x^{13} + 49 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 6$	undefined
7.1.14.26a1.20	42	$x^{14} + 42 x^{13} + 98 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 6$	undefined
7.1.14.26a1.21	42	$x^{14} + 42 x^{13} + 147 x + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 6$	undefined
7.1.14.26a1.22	42	$x^{14} + 7 x^{13} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 5$	undefined
7.1.14.26a1.23	42	$x^{14} + 7 x^{13} + 49 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 5$	undefined
7.1.14.26a1.24	42	$x^{14} + 7 x^{13} + 98 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 5$	undefined
7.1.14.26a1.25	42	$x^{14} + 7 x^{13} + 147 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 5$	undefined
7.1.14.26a1.26	42	$x^{14} + 14 x^{13} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 3$	undefined
7.1.14.26a1.27	42	$x^{14} + 14 x^{13} + 49 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 3$	undefined
7.1.14.26a1.28	42	$x^{14} + 14 x^{13} + 98 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 3$	undefined
7.1.14.26a1.29	42	$x^{14} + 14 x^{13} + 147 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 3$	undefined
7.1.14.26a1.30	42	$x^{14} + 21 x^{13} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 1$	undefined
7.1.14.26a1.31	42	$x^{14} + 21 x^{13} + 49 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 1$	undefined
7.1.14.26a1.32	42	$x^{14} + 21 x^{13} + 98 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 1$	undefined
7.1.14.26a1.33	42	$x^{14} + 21 x^{13} + 147 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 1$	undefined
7.1.14.26a1.34	42	$x^{14} + 28 x^{13} + 49 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 6$	undefined
7.1.14.26a1.35	42	$x^{14} + 28 x^{13} + 98 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 6$	undefined
7.1.14.26a1.36	42	$x^{14} + 28 x^{13} + 147 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 6$	undefined
7.1.14.26a1.37	42	$x^{14} + 35 x^{13} + 49 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 4$	undefined
7.1.14.26a1.38	42	$x^{14} + 35 x^{13} + 98 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 4$	undefined
7.1.14.26a1.39	42	$x^{14} + 35 x^{13} + 147 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 4$	undefined
7.1.14.26a1.40	42	$x^{14} + 42 x^{13} + 49 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 2$	undefined
7.1.14.26a1.41	42	$x^{14} + 42 x^{13} + 98 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 2$	undefined
7.1.14.26a1.42	42	$x^{14} + 42 x^{13} + 147 x + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{25}{12}, \frac{25}{12}]_{12}^{2}$	$[\frac{13}{12},\frac{13}{12}]_{12}^{2}$	$[\frac{25}{12}]^{2}_{6}$	$[\frac{13}{12}]^{2}_{6}$	$[13, 0]$	$[1, 1]$	$z^7 + 2,2 z + 2$	undefined

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