$p$-adic family 2.1.14.16a

• Label: 2.1.14.16a
• Base: 2.1.1.0a1.1
• Degree: \(14\)
• e: \(14\)
• f: \(1\)
• c: \(16\)

Defining polynomial

$x^{14} + 2 c_{6} x^{6} + 2 b_{5} x^{5} + 2 a_{3} x^{3} + 2$

Invariants

Residue field characteristic:	$2$
Degree:	$14$
Base field:	$\Q_{2}$
Ramification index $e$:	$14$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$16$
Artin slopes:	$[\frac{10}{7}]$
Swan slopes:	$[\frac{3}{7}]$
Means:	$\langle\frac{3}{14}\rangle$
Rams:	$(3)$
Field count:	$4$ (complete)
Ambiguity:	$2$
Mass:	$2$
Absolute Mass:	$2$

Diagrams

Varying

Indices of inseparability:	$[3,0]$
Associated inertia:	$[3,1]$
Jump Set:	$[7,17]$

Galois groups and Hidden Artin slopes

Select desired size of Galois group.

		Galois groups of order 168
		$F_8:C_3$ (as 14T11)
hidden slopes	$[\frac{10}{7},\frac{10}{7}]^{3}$	1

		Galois groups of order 336
		$F_8:C_6$ (as 14T18)
hidden slopes	$[\frac{10}{7},\frac{10}{7}]^{6}$	1

		Galois groups of order 1344
		$C_2^3:F_8:C_3$ (as 14T35)
hidden slopes	$[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{10}{7},\frac{10}{7}]^{3}$	1

		Galois groups of order 2688
		$C_2\wr C_7:C_3$ (as 14T44)
hidden slopes	$[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{10}{7},\frac{10}{7}]^{6}$	1

Fields

Showing all 4

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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
2.1.14.16a1.1	1	$x^{14} + 2 x^{3} + 2$	$F_8:C_3$ (as 14T11)	$168$	$2$	$[\frac{10}{7}, \frac{10}{7}, \frac{10}{7}]_{7}^{3}$	$[\frac{3}{7},\frac{3}{7},\frac{3}{7}]_{7}^{3}$	$[\frac{10}{7},\frac{10}{7}]^{3}$	$[\frac{3}{7},\frac{3}{7}]^{3}$	$[3, 0]$	$[3, 1]$	$z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$	$[7, 17]$
2.1.14.16a1.2	1	$x^{14} + 2 x^{6} + 2 x^{3} + 2$	$F_8:C_6$ (as 14T18)	$336$	$2$	$[\frac{10}{7}, \frac{10}{7}, \frac{10}{7}]_{7}^{6}$	$[\frac{3}{7},\frac{3}{7},\frac{3}{7}]_{7}^{6}$	$[\frac{10}{7},\frac{10}{7}]^{6}$	$[\frac{3}{7},\frac{3}{7}]^{6}$	$[3, 0]$	$[3, 1]$	$z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$	$[7, 17]$
2.1.14.16a1.3	1	$x^{14} + 2 x^{5} + 2 x^{3} + 2$	$C_2^3:F_8:C_3$ (as 14T35)	$1344$	$2$	$[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, \frac{10}{7}, \frac{10}{7}, \frac{10}{7}]_{7}^{3}$	$[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{3}{7},\frac{3}{7},\frac{3}{7}]_{7}^{3}$	$[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{10}{7},\frac{10}{7}]^{3}$	$[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{3}{7},\frac{3}{7}]^{3}$	$[3, 0]$	$[3, 1]$	$z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$	$[7, 17]$
2.1.14.16a1.4	1	$x^{14} + 2 x^{6} + 2 x^{5} + 2 x^{3} + 2$	$C_2\wr C_7:C_3$ (as 14T44)	$2688$	$2$	$[\frac{8}{7}, \frac{8}{7}, \frac{8}{7}, \frac{10}{7}, \frac{10}{7}, \frac{10}{7}]_{7}^{6}$	$[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{3}{7},\frac{3}{7},\frac{3}{7}]_{7}^{6}$	$[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{10}{7},\frac{10}{7}]^{6}$	$[\frac{1}{7},\frac{1}{7},\frac{1}{7},\frac{3}{7},\frac{3}{7}]^{6}$	$[3, 0]$	$[3, 1]$	$z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$	$[7, 17]$

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