$p$-adic family 2.1.8.24c1.56-1.2.4a

• Label: 2.1.8.24c1.56-1.2.4a
• Base: 2.1.8.24c1.56
• 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:	2.1.8.24c1.56
Ramification index $e$:	$2$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$4$
Absolute Artin slopes:	$[2,3,3,4]$
Swan slopes:	$[3]$
Means:	$\langle\frac{3}{2}\rangle$
Rams:	$(3)$
Field count:	$2$ (complete)
Ambiguity:	$2$
Mass:	$2$
Absolute Mass:	$1/4$

Diagrams

Varying

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

Galois group:	$C_2^2:Q_8$
Hidden Artin slopes:	$[\ ]^{2}$
Indices of inseparability:	$[37,26,24,8,0]$
Associated inertia:	$[1,2,1]$
Jump Set:	$[1,2,4,8,32]$

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
2.1.16.52k1.244	$x^{16} + 4 x^{14} + 8 x^{11} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 30$	$C_2^2:Q_8$ (as 16T31)	$32$	$8$	$[2, 3, 3, 4]^{2}$	$[1,2,2,3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[37, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 2, 4, 8, 32]$
2.1.16.52k1.276	$x^{16} + 8 x^{15} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{5} + 14$	$C_2^2:Q_8$ (as 16T31)	$32$	$8$	$[2, 3, 3, 4]^{2}$	$[1,2,2,3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[37, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 2, 4, 8, 32]$

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