$p$-adic family 3.1.6.11a

• Label: 3.1.6.11a
• Base: 3.1.1.0a1.1
• Degree: \(6\)
• e: \(6\)
• f: \(1\)
• c: \(11\)

Defining polynomial

$x^{6} + 9 c_{9} x^{3} + 9 b_{8} x^{2} + 9 b_{7} x + 3 d_{0}$

Invariants

Residue field characteristic:	$3$
Degree:	$6$
Base field:	$\Q_{3}$
Ramification index $e$:	$6$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$11$
Artin slopes:	$[\frac{5}{2}]$
Swan slopes:	$[\frac{3}{2}]$
Means:	$\langle1\rangle$
Rams:	$(3)$
Field count:	$21$ (complete)
Ambiguity:	$6$
Mass:	$9$
Absolute Mass:	$9$

Diagrams

Varying

Indices of inseparability:	$[6,0]$
Associated inertia:	$[1,1]$ (show 15), $[1,2]$ (show 6)
Jump Set:	undefined (show 6), $[1,7]$ (show 15)

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 6
		$S_3$ (as 6T2)
hidden slopes	$[\ ]$	3

		Galois groups of order 12
		$D_6$ (as 6T3)
hidden slopes	$[\ ]^{2}$	3

		Galois groups of order 18
		$C_3\times S_3$ (as 6T5)
hidden slopes	$[2]$	9
		$[\ ]^{3}$	3
	Total		12

		Galois groups of order 36
		$S_3^2$ (as 6T9)
hidden slopes	$[2]^{2}$	3

Fields

Showing all 21

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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
3.1.6.11a1.1	3	$x^{6} + 3$	$S_3$ (as 6T2)	$6$	$6$	$[\frac{5}{2}]_{2}$	$[\frac{3}{2}]_{2}$	$[\ ]$	$[\ ]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 7]$
3.1.6.11a1.2	3	$x^{6} + 9 x^{3} + 3$	$S_3\times C_3$ (as 6T5)	$18$	$3$	$[\frac{5}{2}]_{2}^{3}$	$[\frac{3}{2}]_{2}^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 7]$
3.1.6.11a1.3	3	$x^{6} + 9 x^{2} + 3$	$S_3$ (as 6T2)	$6$	$6$	$[\frac{5}{2}]_{2}$	$[\frac{3}{2}]_{2}$	$[\ ]$	$[\ ]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 7]$
3.1.6.11a1.4	3	$x^{6} + 9 x^{3} + 9 x^{2} + 3$	$S_3\times C_3$ (as 6T5)	$18$	$3$	$[\frac{5}{2}]_{2}^{3}$	$[\frac{3}{2}]_{2}^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 7]$
3.1.6.11a1.5	3	$x^{6} + 18 x^{2} + 3$	$S_3$ (as 6T2)	$6$	$6$	$[\frac{5}{2}]_{2}$	$[\frac{3}{2}]_{2}$	$[\ ]$	$[\ ]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 7]$
3.1.6.11a1.6	3	$x^{6} + 9 x^{3} + 18 x^{2} + 3$	$S_3\times C_3$ (as 6T5)	$18$	$3$	$[\frac{5}{2}]_{2}^{3}$	$[\frac{3}{2}]_{2}^{3}$	$[\ ]^{3}$	$[\ ]^{3}$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 7]$
3.1.6.11a1.7	9	$x^{6} + 9 x + 3$	$S_3\times C_3$ (as 6T5)	$18$	$3$	$[2, \frac{5}{2}]_{2}$	$[1,\frac{3}{2}]_{2}$	$[2]$	$[1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 7]$
3.1.6.11a1.8	9	$x^{6} + 9 x^{3} + 9 x + 3$	$S_3\times C_3$ (as 6T5)	$18$	$3$	$[2, \frac{5}{2}]_{2}$	$[1,\frac{3}{2}]_{2}$	$[2]$	$[1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 7]$
3.1.6.11a1.9	9	$x^{6} + 18 x^{3} + 9 x + 3$	$S_3\times C_3$ (as 6T5)	$18$	$3$	$[2, \frac{5}{2}]_{2}$	$[1,\frac{3}{2}]_{2}$	$[2]$	$[1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 7]$
3.1.6.11a1.10	9	$x^{6} + 9 x^{2} + 9 x + 3$	$S_3\times C_3$ (as 6T5)	$18$	$3$	$[2, \frac{5}{2}]_{2}$	$[1,\frac{3}{2}]_{2}$	$[2]$	$[1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 7]$
3.1.6.11a1.11	9	$x^{6} + 9 x^{3} + 9 x^{2} + 9 x + 3$	$S_3\times C_3$ (as 6T5)	$18$	$3$	$[2, \frac{5}{2}]_{2}$	$[1,\frac{3}{2}]_{2}$	$[2]$	$[1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 7]$
3.1.6.11a1.12	9	$x^{6} + 18 x^{3} + 9 x^{2} + 9 x + 3$	$S_3\times C_3$ (as 6T5)	$18$	$3$	$[2, \frac{5}{2}]_{2}$	$[1,\frac{3}{2}]_{2}$	$[2]$	$[1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 7]$
3.1.6.11a1.13	9	$x^{6} + 18 x^{2} + 9 x + 3$	$S_3\times C_3$ (as 6T5)	$18$	$3$	$[2, \frac{5}{2}]_{2}$	$[1,\frac{3}{2}]_{2}$	$[2]$	$[1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 7]$
3.1.6.11a1.14	9	$x^{6} + 9 x^{3} + 18 x^{2} + 9 x + 3$	$S_3\times C_3$ (as 6T5)	$18$	$3$	$[2, \frac{5}{2}]_{2}$	$[1,\frac{3}{2}]_{2}$	$[2]$	$[1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 7]$
3.1.6.11a1.15	9	$x^{6} + 18 x^{3} + 18 x^{2} + 9 x + 3$	$S_3\times C_3$ (as 6T5)	$18$	$3$	$[2, \frac{5}{2}]_{2}$	$[1,\frac{3}{2}]_{2}$	$[2]$	$[1]$	$[6, 0]$	$[1, 1]$	$z^3 + 2,2 z^2 + 1$	$[1, 7]$
3.1.6.11a2.1	3	$x^{6} + 6$	$D_{6}$ (as 6T3)	$12$	$2$	$[\frac{5}{2}]_{2}^{2}$	$[\frac{3}{2}]_{2}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + 2$	undefined
3.1.6.11a2.2	3	$x^{6} + 9 x^{2} + 6$	$D_{6}$ (as 6T3)	$12$	$2$	$[\frac{5}{2}]_{2}^{2}$	$[\frac{3}{2}]_{2}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + 2$	undefined
3.1.6.11a2.3	3	$x^{6} + 18 x^{2} + 6$	$D_{6}$ (as 6T3)	$12$	$2$	$[\frac{5}{2}]_{2}^{2}$	$[\frac{3}{2}]_{2}^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + 2$	undefined
3.1.6.11a2.4	3	$x^{6} + 9 x + 6$	$S_3^2$ (as 6T9)	$36$	$1$	$[2, \frac{5}{2}]_{2}^{2}$	$[1,\frac{3}{2}]_{2}^{2}$	$[2]^{2}$	$[1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + 2$	undefined
3.1.6.11a2.5	3	$x^{6} + 9 x^{2} + 9 x + 6$	$S_3^2$ (as 6T9)	$36$	$1$	$[2, \frac{5}{2}]_{2}^{2}$	$[1,\frac{3}{2}]_{2}^{2}$	$[2]^{2}$	$[1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + 2$	undefined
3.1.6.11a2.6	3	$x^{6} + 18 x^{2} + 9 x + 6$	$S_3^2$ (as 6T9)	$36$	$1$	$[2, \frac{5}{2}]_{2}^{2}$	$[1,\frac{3}{2}]_{2}^{2}$	$[2]^{2}$	$[1]^{2}$	$[6, 0]$	$[1, 2]$	$z^3 + 2,2 z^2 + 2$	undefined

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