$p$-adic family 2.1.7.6a1.1-1.2.8a

• Label: 2.1.7.6a1.1-1.2.8a
• Base: 2.1.7.6a1.1
• Degree: \(2\)
• e: \(2\)
• f: \(1\)
• c: \(8\)

Defining polynomial

$x^{2} + \left(b_{13} \pi^{7} + b_{11} \pi^{6} + b_{9} \pi^{5} + a_{7} \pi^{4}\right) x + c_{14} \pi^{8} + \pi$

Invariants

Residue field characteristic:	$2$
Degree:	$2$
Base field:	2.1.7.6a1.1
Ramification index $e$:	$2$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$8$
Absolute Artin slopes:	$[2]$
Swan slopes:	$[7]$
Means:	$\langle\frac{7}{2}\rangle$
Rams:	$(7)$
Field count:	$16$ (complete)
Ambiguity:	$2$
Mass:	$8$
Absolute Mass:	$8$

Diagrams

Varying

These invariants are all associated to absolute extensions of $\Q_{ 2 }$ within this relative family, not the relative extension.

Galois group:	$(C_7:C_3) \times C_2$ (show 2), $F_8:C_6$ (show 8), $C_2\wr C_7:C_3$ (show 6)
Hidden Artin slopes:	$[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$ (show 4), $[\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$ (show 4), $[\frac{8}{7},\frac{8}{7},\frac{8}{7}]^{3}$ (show 2), $[\frac{10}{7},\frac{10}{7},\frac{10}{7}]^{3}$ (show 2), $[\ ]^{3}$ (show 2), $[\frac{8}{7},\frac{8}{7},\frac{8}{7},\frac{10}{7},\frac{10}{7},\frac{10}{7}]^{3}$ (show 2)
Indices of inseparability:	$[7,0]$
Associated inertia:	$[3,1]$
Jump Set:	$[7,14]$ (show 1), $[7,23]$ (show 8), $[7,25]$ (show 4), $[7,27]$ (show 2), $[7,28]$ (show 1)

Fields

Showing all 4

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Label	Polynomial $/ \Q_p$	Galois group $/ \Q_p$	Galois degree $/ \Q_p$	$\#\Aut(K/\Q_p)$	Artin slope content $/ \Q_p$	Swan slope content $/ \Q_p$	Hidden Artin slopes $/ \Q_p$	Hidden Swan slopes $/ \Q_p$	Ind. of Insep. $/ \Q_p$	Assoc. Inertia $/ \Q_p$	Resid. Poly	Jump Set
2.1.14.20a1.11	$x^{14} + 2 x^{13} + 2 x^{9} + 2 x^{7} + 2$	$F_8:C_6$ (as 14T18)	$336$	$2$	$[\frac{12}{7}, \frac{12}{7}, \frac{12}{7}, 2]_{7}^{3}$	$[\frac{5}{7},\frac{5}{7},\frac{5}{7},1]_{7}^{3}$	$[\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$	$[\frac{5}{7},\frac{5}{7},\frac{5}{7}]^{3}$	$[7, 0]$	$[3, 1]$	$z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$	$[7, 23]$
2.1.14.20a1.12	$x^{14} + 2 x^{13} + 2 x^{9} + 2 x^{7} + 6$	$F_8:C_6$ (as 14T18)	$336$	$2$	$[\frac{12}{7}, \frac{12}{7}, \frac{12}{7}, 2]_{7}^{3}$	$[\frac{5}{7},\frac{5}{7},\frac{5}{7},1]_{7}^{3}$	$[\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$	$[\frac{5}{7},\frac{5}{7},\frac{5}{7}]^{3}$	$[7, 0]$	$[3, 1]$	$z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$	$[7, 23]$
2.1.14.20a1.15	$x^{14} + 2 x^{13} + 2 x^{11} + 2 x^{9} + 2 x^{7} + 2$	$F_8:C_6$ (as 14T18)	$336$	$2$	$[\frac{12}{7}, \frac{12}{7}, \frac{12}{7}, 2]_{7}^{3}$	$[\frac{5}{7},\frac{5}{7},\frac{5}{7},1]_{7}^{3}$	$[\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$	$[\frac{5}{7},\frac{5}{7},\frac{5}{7}]^{3}$	$[7, 0]$	$[3, 1]$	$z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$	$[7, 23]$
2.1.14.20a1.16	$x^{14} + 2 x^{13} + 2 x^{11} + 2 x^{9} + 2 x^{7} + 6$	$F_8:C_6$ (as 14T18)	$336$	$2$	$[\frac{12}{7}, \frac{12}{7}, \frac{12}{7}, 2]_{7}^{3}$	$[\frac{5}{7},\frac{5}{7},\frac{5}{7},1]_{7}^{3}$	$[\frac{12}{7},\frac{12}{7},\frac{12}{7}]^{3}$	$[\frac{5}{7},\frac{5}{7},\frac{5}{7}]^{3}$	$[7, 0]$	$[3, 1]$	$z^{12} + z^{10} + z^8 + z^6 + z^4 + z^2 + 1,z + 1$	$[7, 23]$

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