$p$-adic family 2.1.8.24c1.56-1.2.6a

• Label: 2.1.8.24c1.56-1.2.6a
• Base: 2.1.8.24c1.56
• Degree: \(2\)
• e: \(2\)
• f: \(1\)
• c: \(6\)

Defining polynomial

$x^{2} + \left(b_{9} \pi^{5} + b_{7} \pi^{4} + a_{5} \pi^{3}\right) x + c_{10} \pi^{6} + \pi$

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 $c$:	$6$
Absolute Artin slopes:	$[2,3,\frac{7}{2},4]$
Swan slopes:	$[5]$
Means:	$\langle\frac{5}{2}\rangle$
Rams:	$(5)$
Field count:	$5$ (complete)
Ambiguity:	$2$
Mass:	$4$
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:	$C_4:C_4$ (show 2), $C_2^2:Q_8$ (show 3)
Hidden Artin slopes:	$[\ ]$ (show 2), $[\ ]^{2}$ (show 3)
Indices of inseparability:	$[39,30,20,8,0]$
Associated inertia:	$[1,1,1,1]$
Jump Set:	$[1,2,4,8,32]$

Fields

Showing all 5

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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
2.1.16.54o1.286	$x^{16} + 4 x^{14} + 8 x^{13} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 4 x^{4} + 14$	$C_2^2:Q_8$ (as 16T31)	$32$	$8$	$[2, 3, \frac{7}{2}, 4]^{2}$	$[1,2,\frac{5}{2},3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[39, 30, 20, 8, 0]$	$[1, 1, 1, 1]$	$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 4, 8, 32]$
2.1.16.54o1.297	$x^{16} + 8 x^{15} + 4 x^{14} + 8 x^{13} + 8 x^{9} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 14$	$C_4:C_4$ (as 16T8)	$16$	$16$	$[2, 3, \frac{7}{2}, 4]$	$[1,2,\frac{5}{2},3]$	$[\ ]$	$[\ ]$	$[39, 30, 20, 8, 0]$	$[1, 1, 1, 1]$	$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 4, 8, 32]$
2.1.16.54o1.307	$x^{16} + 4 x^{14} + 8 x^{13} + 8 x^{9} + 10 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 30$	$C_4:C_4$ (as 16T8)	$16$	$16$	$[2, 3, \frac{7}{2}, 4]$	$[1,2,\frac{5}{2},3]$	$[\ ]$	$[\ ]$	$[39, 30, 20, 8, 0]$	$[1, 1, 1, 1]$	$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 4, 8, 32]$
2.1.16.54o1.315	$x^{16} + 4 x^{14} + 8 x^{11} + 2 x^{8} + 8 x^{7} + 4 x^{4} + 8 x^{2} + 14$	$C_2^2:Q_8$ (as 16T31)	$32$	$8$	$[2, 3, \frac{7}{2}, 4]^{2}$	$[1,2,\frac{5}{2},3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[39, 30, 20, 8, 0]$	$[1, 1, 1, 1]$	$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 4, 8, 32]$
2.1.16.54o1.326	$x^{16} + 8 x^{15} + 4 x^{14} + 2 x^{8} + 8 x^{7} + 8 x^{6} + 4 x^{4} + 8 x^{2} + 14$	$C_2^2:Q_8$ (as 16T31)	$32$	$8$	$[2, 3, \frac{7}{2}, 4]^{2}$	$[1,2,\frac{5}{2},3]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[39, 30, 20, 8, 0]$	$[1, 1, 1, 1]$	$z^8 + 1,z^4 + 1,z^2 + 1,z + 1$	$[1, 2, 4, 8, 32]$

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