$p$-adic family 2.1.4.11a1.12-1.2.4a

• Label: 2.1.4.11a1.12-1.2.4a
• Base: 2.1.4.11a1.12
• Degree: \(2\)
• e: \(2\)
• f: \(1\)
• c: \(4\)

Defining polynomial

$x^{2} + \left(b_{5} \pi^{3} + a_{3} \pi^{2}\right) x + c_{6} \pi^{4} + \pi$

Invariants

Residue field characteristic:	$2$
Degree:	$2$
Base field:	2.1.4.11a1.12
Ramification index $e$:	$2$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$4$
Absolute Artin slopes:	$[3,\frac{7}{2},4]$
Swan slopes:	$[3]$
Means:	$\langle\frac{3}{2}\rangle$
Rams:	$(3)$
Field count:	$2$ (complete)
Ambiguity:	$2$
Mass:	$2$
Absolute Mass:	$1/2$

Diagrams

Varying

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

Galois group:	$C_2^2:C_4$
Hidden Artin slopes:	$[2]$
Indices of inseparability:	$[19,14,8,0]$
Associated inertia:	$[1,1,1]$
Jump Set:	$[1,3,7,15]$

Fields

Showing all 2

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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
2.1.8.26c1.6	$x^{8} + 8 x^{7} + 4 x^{6} + 8 x^{5} + 8 x^{3} + 18$	$C_2^2:C_4$ (as 8T10)	$16$	$4$	$[2, 3, \frac{7}{2}, 4]$	$[1,2,\frac{5}{2},3]$	$[2]$	$[1]$	$[19, 14, 8, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 3, 7, 15]$
2.1.8.26c1.10	$x^{8} + 4 x^{6} + 8 x^{5} + 8 x^{4} + 8 x^{3} + 18$	$C_2^2:C_4$ (as 8T10)	$16$	$4$	$[2, 3, \frac{7}{2}, 4]$	$[1,2,\frac{5}{2},3]$	$[2]$	$[1]$	$[19, 14, 8, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 3, 7, 15]$

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