$p$-adic family 2.1.8.16c

• Label: 2.1.8.16c
• Base: 2.1.1.0a1.1
• Degree: \(8\)
• e: \(8\)
• f: \(1\)
• c: \(16\)

Defining polynomial

$x^{8} + 2 a_{6} x^{6} + \left(2 b_{4} + 4 c_{12}\right) x^{4} + 4 b_{11} x^{3} + 4 a_{9} x + 4 c_{8} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$8$
Base field:	$\Q_{2}$
Ramification index $e$:	$8$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$16$
Artin slopes:	$[2,2,\frac{5}{2}]$
Swan slopes:	$[1,1,\frac{3}{2}]$
Means:	$\langle\frac{1}{2},\frac{3}{4},\frac{9}{8}\rangle$
Rams:	$(1,1,3)$
Field count:	$10$ (complete)
Ambiguity:	$4$
Mass:	$4$
Absolute Mass:	$4$

Diagrams

Varying

Indices of inseparability:	$[9,6,4,0]$ (show 4), $[9,6,6,0]$ (show 6)
Associated inertia:	$[2,1]$ (show 6), $[3,1]$ (show 4)
Jump Set:	$[1,2,7,15]$ (show 4), $[1,3,6,16]$ (show 3), $[1,3,9,17]$ (show 3)

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 16
		$\SD_{16}$ (as 8T8)
hidden slopes	$[\ ]^{2}$	2

		Galois groups of order 32
		$C_4\wr C_2$ (as 8T17)
hidden slopes	$[\ ]^{4}$	4

		Galois groups of order 192
		$C_2\wr A_4$ (as 8T38)
hidden slopes	$[2,2]^{6}$	4

Fields

Showing all 6

  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
2.1.8.16c1.1	4	$x^{8} + 2 x^{6} + 4 x + 2$	$C_4\wr C_2$ (as 8T17)	$32$	$4$	$[2, 2, \frac{5}{2}]^{4}$	$[1,1,\frac{3}{2}]^{4}$	$[\ ]^{4}$	$[\ ]^{4}$	$[9, 6, 6, 0]$	$[2, 1]$	$z^6 + 1,z + 1$	$[1, 3, 9, 17]$
2.1.8.16c1.2	4	$x^{8} + 2 x^{6} + 4 x^{4} + 4 x + 2$	$C_4\wr C_2$ (as 8T17)	$32$	$4$	$[2, 2, \frac{5}{2}]^{4}$	$[1,1,\frac{3}{2}]^{4}$	$[\ ]^{4}$	$[\ ]^{4}$	$[9, 6, 6, 0]$	$[2, 1]$	$z^6 + 1,z + 1$	$[1, 3, 9, 17]$
2.1.8.16c1.3	2	$x^{8} + 2 x^{6} + 4 x^{3} + 4 x + 2$	$QD_{16}$ (as 8T8)	$16$	$2$	$[2, 2, \frac{5}{2}]^{2}$	$[1,1,\frac{3}{2}]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[9, 6, 6, 0]$	$[2, 1]$	$z^6 + 1,z + 1$	$[1, 3, 9, 17]$
2.1.8.16c1.4	4	$x^{8} + 2 x^{6} + 4 x + 6$	$C_4\wr C_2$ (as 8T17)	$32$	$4$	$[2, 2, \frac{5}{2}]^{4}$	$[1,1,\frac{3}{2}]^{4}$	$[\ ]^{4}$	$[\ ]^{4}$	$[9, 6, 6, 0]$	$[2, 1]$	$z^6 + 1,z + 1$	$[1, 3, 6, 16]$
2.1.8.16c1.5	4	$x^{8} + 2 x^{6} + 4 x^{4} + 4 x + 6$	$C_4\wr C_2$ (as 8T17)	$32$	$4$	$[2, 2, \frac{5}{2}]^{4}$	$[1,1,\frac{3}{2}]^{4}$	$[\ ]^{4}$	$[\ ]^{4}$	$[9, 6, 6, 0]$	$[2, 1]$	$z^6 + 1,z + 1$	$[1, 3, 6, 16]$
2.1.8.16c1.6	2	$x^{8} + 2 x^{6} + 4 x^{3} + 4 x + 6$	$QD_{16}$ (as 8T8)	$16$	$2$	$[2, 2, \frac{5}{2}]^{2}$	$[1,1,\frac{3}{2}]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[9, 6, 6, 0]$	$[2, 1]$	$z^6 + 1,z + 1$	$[1, 3, 6, 16]$

  Download displayed columns for results to