$p$-adic family 7.1.14.22a

• Label: 7.1.14.22a
• Base: 7.1.1.0a1.1
• Degree: \(14\)
• e: \(14\)
• f: \(1\)
• c: \(22\)

Defining polynomial

$x^{14} + 7 b_{10} x^{10} + 7 a_{9} x^{9} + 7 d_{0}$

Invariants

Residue field characteristic:	$7$
Degree:	$14$
Base field:	$\Q_{7}$
Ramification index $e$:	$14$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$22$
Artin slopes:	$[\frac{7}{4}]$
Swan slopes:	$[\frac{3}{4}]$
Means:	$\langle\frac{9}{14}\rangle$
Rams:	$(\frac{3}{2})$
Field count:	$42$ (complete)
Ambiguity:	$2$
Mass:	$42$
Absolute Mass:	$42$

Diagrams

Varying

Indices of inseparability:	$[9,0]$
Associated inertia:	$[1,1]$ (show 14), $[1,3]$ (show 28)
Jump Set:	undefined

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 392
		$D_7\wr C_2$ (as 14T20)
hidden slopes	$[\frac{7}{4}]^{2}_{2}$	14

		Galois groups of order 1176
		$D_7^2:C_6$ (as 14T32)
hidden slopes	$[\frac{7}{4}]^{6}_{2}$	28

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
7.1.14.22a1.1	14	$x^{14} + 21 x^{9} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 1$	undefined
7.1.14.22a1.2	14	$x^{14} + 7 x^{10} + 21 x^{9} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 1$	undefined
7.1.14.22a1.3	14	$x^{14} + 14 x^{10} + 21 x^{9} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 1$	undefined
7.1.14.22a1.4	14	$x^{14} + 21 x^{10} + 21 x^{9} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 1$	undefined
7.1.14.22a1.5	14	$x^{14} + 28 x^{10} + 21 x^{9} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 1$	undefined
7.1.14.22a1.6	14	$x^{14} + 35 x^{10} + 21 x^{9} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 1$	undefined
7.1.14.22a1.7	14	$x^{14} + 42 x^{10} + 21 x^{9} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 1$	undefined
7.1.14.22a1.8	14	$x^{14} + 14 x^{9} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 1$	undefined
7.1.14.22a1.9	14	$x^{14} + 7 x^{10} + 14 x^{9} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 1$	undefined
7.1.14.22a1.10	14	$x^{14} + 14 x^{10} + 14 x^{9} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 1$	undefined
7.1.14.22a1.11	14	$x^{14} + 21 x^{10} + 14 x^{9} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 1$	undefined
7.1.14.22a1.12	14	$x^{14} + 28 x^{10} + 14 x^{9} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 1$	undefined
7.1.14.22a1.13	14	$x^{14} + 35 x^{10} + 14 x^{9} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 1$	undefined
7.1.14.22a1.14	14	$x^{14} + 42 x^{10} + 14 x^{9} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 1$	undefined
7.1.14.22a2.1	14	$x^{14} + 7 x^{9} + 7$	$D_7 \wr C_2$ (as 14T20)	$392$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{7}{4}]^{2}_{2}$	$[\frac{3}{4}]^{2}_{2}$	$[9, 0]$	$[1, 1]$	$z^7 + 2,2 z^3 + 5$	undefined
7.1.14.22a2.2	14	$x^{14} + 7 x^{10} + 7 x^{9} + 7$	$D_7 \wr C_2$ (as 14T20)	$392$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{7}{4}]^{2}_{2}$	$[\frac{3}{4}]^{2}_{2}$	$[9, 0]$	$[1, 1]$	$z^7 + 2,2 z^3 + 5$	undefined
7.1.14.22a2.3	14	$x^{14} + 14 x^{10} + 7 x^{9} + 7$	$D_7 \wr C_2$ (as 14T20)	$392$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{7}{4}]^{2}_{2}$	$[\frac{3}{4}]^{2}_{2}$	$[9, 0]$	$[1, 1]$	$z^7 + 2,2 z^3 + 5$	undefined
7.1.14.22a2.4	14	$x^{14} + 21 x^{10} + 7 x^{9} + 7$	$D_7 \wr C_2$ (as 14T20)	$392$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{7}{4}]^{2}_{2}$	$[\frac{3}{4}]^{2}_{2}$	$[9, 0]$	$[1, 1]$	$z^7 + 2,2 z^3 + 5$	undefined
7.1.14.22a2.5	14	$x^{14} + 28 x^{10} + 7 x^{9} + 7$	$D_7 \wr C_2$ (as 14T20)	$392$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{7}{4}]^{2}_{2}$	$[\frac{3}{4}]^{2}_{2}$	$[9, 0]$	$[1, 1]$	$z^7 + 2,2 z^3 + 5$	undefined
7.1.14.22a2.6	14	$x^{14} + 35 x^{10} + 7 x^{9} + 7$	$D_7 \wr C_2$ (as 14T20)	$392$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{7}{4}]^{2}_{2}$	$[\frac{3}{4}]^{2}_{2}$	$[9, 0]$	$[1, 1]$	$z^7 + 2,2 z^3 + 5$	undefined
7.1.14.22a2.7	14	$x^{14} + 42 x^{10} + 7 x^{9} + 7$	$D_7 \wr C_2$ (as 14T20)	$392$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{7}{4}]^{2}_{2}$	$[\frac{3}{4}]^{2}_{2}$	$[9, 0]$	$[1, 1]$	$z^7 + 2,2 z^3 + 5$	undefined
7.1.14.22a2.8	14	$x^{14} + 21 x^{9} + 21$	$D_7 \wr C_2$ (as 14T20)	$392$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{7}{4}]^{2}_{2}$	$[\frac{3}{4}]^{2}_{2}$	$[9, 0]$	$[1, 1]$	$z^7 + 2,2 z^3 + 5$	undefined
7.1.14.22a2.9	14	$x^{14} + 7 x^{10} + 21 x^{9} + 21$	$D_7 \wr C_2$ (as 14T20)	$392$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{7}{4}]^{2}_{2}$	$[\frac{3}{4}]^{2}_{2}$	$[9, 0]$	$[1, 1]$	$z^7 + 2,2 z^3 + 5$	undefined
7.1.14.22a2.10	14	$x^{14} + 14 x^{10} + 21 x^{9} + 21$	$D_7 \wr C_2$ (as 14T20)	$392$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{7}{4}]^{2}_{2}$	$[\frac{3}{4}]^{2}_{2}$	$[9, 0]$	$[1, 1]$	$z^7 + 2,2 z^3 + 5$	undefined
7.1.14.22a2.11	14	$x^{14} + 21 x^{10} + 21 x^{9} + 21$	$D_7 \wr C_2$ (as 14T20)	$392$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{7}{4}]^{2}_{2}$	$[\frac{3}{4}]^{2}_{2}$	$[9, 0]$	$[1, 1]$	$z^7 + 2,2 z^3 + 5$	undefined
7.1.14.22a2.12	14	$x^{14} + 28 x^{10} + 21 x^{9} + 21$	$D_7 \wr C_2$ (as 14T20)	$392$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{7}{4}]^{2}_{2}$	$[\frac{3}{4}]^{2}_{2}$	$[9, 0]$	$[1, 1]$	$z^7 + 2,2 z^3 + 5$	undefined
7.1.14.22a2.13	14	$x^{14} + 35 x^{10} + 21 x^{9} + 21$	$D_7 \wr C_2$ (as 14T20)	$392$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{7}{4}]^{2}_{2}$	$[\frac{3}{4}]^{2}_{2}$	$[9, 0]$	$[1, 1]$	$z^7 + 2,2 z^3 + 5$	undefined
7.1.14.22a2.14	14	$x^{14} + 42 x^{10} + 21 x^{9} + 21$	$D_7 \wr C_2$ (as 14T20)	$392$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{2}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{2}$	$[\frac{7}{4}]^{2}_{2}$	$[\frac{3}{4}]^{2}_{2}$	$[9, 0]$	$[1, 1]$	$z^7 + 2,2 z^3 + 5$	undefined
7.1.14.22a3.1	14	$x^{14} + 14 x^{9} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 3$	undefined
7.1.14.22a3.2	14	$x^{14} + 7 x^{10} + 14 x^{9} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 3$	undefined
7.1.14.22a3.3	14	$x^{14} + 14 x^{10} + 14 x^{9} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 3$	undefined
7.1.14.22a3.4	14	$x^{14} + 21 x^{10} + 14 x^{9} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 3$	undefined
7.1.14.22a3.5	14	$x^{14} + 28 x^{10} + 14 x^{9} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 3$	undefined
7.1.14.22a3.6	14	$x^{14} + 35 x^{10} + 14 x^{9} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 3$	undefined
7.1.14.22a3.7	14	$x^{14} + 42 x^{10} + 14 x^{9} + 7$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 3$	undefined
7.1.14.22a3.8	14	$x^{14} + 7 x^{9} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 4$	undefined
7.1.14.22a3.9	14	$x^{14} + 7 x^{10} + 7 x^{9} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 4$	undefined
7.1.14.22a3.10	14	$x^{14} + 14 x^{10} + 7 x^{9} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 4$	undefined
7.1.14.22a3.11	14	$x^{14} + 21 x^{10} + 7 x^{9} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 4$	undefined
7.1.14.22a3.12	14	$x^{14} + 28 x^{10} + 7 x^{9} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 4$	undefined
7.1.14.22a3.13	14	$x^{14} + 35 x^{10} + 7 x^{9} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 4$	undefined
7.1.14.22a3.14	14	$x^{14} + 42 x^{10} + 7 x^{9} + 21$	$D_7^2:C_6$ (as 14T32)	$1176$	$1$	$[\frac{7}{4}, \frac{7}{4}]_{4}^{6}$	$[\frac{3}{4},\frac{3}{4}]_{4}^{6}$	$[\frac{7}{4}]^{6}_{2}$	$[\frac{3}{4}]^{6}_{2}$	$[9, 0]$	$[1, 3]$	$z^7 + 2,2 z^3 + 4$	undefined

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