$p$-adic family 2.1.4.8b1.5-1.4.12b

• Label: 2.1.4.8b1.5-1.4.12b
• Base: 2.1.4.8b1.5
• Degree: \(4\)
• e: \(4\)
• f: \(1\)
• c: \(12\)

Defining polynomial

$x^{4} + \left(b_{15} \pi^{4} + b_{11} \pi^{3}\right) x^{3} + a_{2} \pi x^{2} + \left(b_{13} \pi^{4} + a_{9} \pi^{3}\right) x + c_{16} \pi^{5} + c_{4} \pi^{2} + \pi$

Invariants

Residue field characteristic:	$2$
Degree:	$4$
Base field:	2.1.4.8b1.5
Ramification index $e$:	$4$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$12$
Absolute Artin slopes:	$[2,2,3,\frac{13}{4}]$
Swan slopes:	$[1,4]$
Means:	$\langle\frac{1}{2},\frac{9}{4}\rangle$
Rams:	$(1,7)$
Field count:	$4$ (complete)
Ambiguity:	$4$
Mass:	$8$
Absolute Mass:	$2$

Diagrams

Varying

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

Galois group:	$C_2^5.(C_2\times D_4)$
Hidden Artin slopes:	$[2,3,3]^{4}$
Indices of inseparability:	$[29,22,12,12,0]$
Associated inertia:	$[2,1,1]$
Jump Set:	$[1,3,6,12,32]$

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
2.1.16.44m1.53	$x^{16} + 4 x^{15} + 4 x^{13} + 2 x^{12} + 4 x^{10} + 4 x^{6} + 6$	$C_2^5.(C_2\times D_4)$ (as 16T813)	$512$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{4}$	$[1,1,1,2,2,2,\frac{9}{4}]^{4}$	$[2,3,3]^{4}$	$[1,2,2]^{4}$	$[29, 22, 12, 12, 0]$	$[2, 1, 1]$	$z^{12} + 1,z^2 + 1,z + 1$	$[1, 3, 6, 12, 32]$
2.1.16.44m1.54	$x^{16} + 4 x^{15} + 4 x^{13} + 2 x^{12} + 4 x^{10} + 4 x^{6} + 8 x^{4} + 6$	$C_2^5.(C_2\times D_4)$ (as 16T813)	$512$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{4}$	$[1,1,1,2,2,2,\frac{9}{4}]^{4}$	$[2,3,3]^{4}$	$[1,2,2]^{4}$	$[29, 22, 12, 12, 0]$	$[2, 1, 1]$	$z^{12} + 1,z^2 + 1,z + 1$	$[1, 3, 6, 12, 32]$
2.1.16.44m1.57	$x^{16} + 4 x^{14} + 4 x^{13} + 2 x^{12} + 4 x^{10} + 4 x^{6} + 6$	$C_2^5.(C_2\times D_4)$ (as 16T813)	$512$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{4}$	$[1,1,1,2,2,2,\frac{9}{4}]^{4}$	$[2,3,3]^{4}$	$[1,2,2]^{4}$	$[29, 22, 12, 12, 0]$	$[2, 1, 1]$	$z^{12} + 1,z^2 + 1,z + 1$	$[1, 3, 6, 12, 32]$
2.1.16.44m1.58	$x^{16} + 4 x^{14} + 4 x^{13} + 2 x^{12} + 4 x^{10} + 4 x^{6} + 8 x^{4} + 6$	$C_2^5.(C_2\times D_4)$ (as 16T813)	$512$	$2$	$[2, 2, 2, 3, 3, 3, \frac{13}{4}]^{4}$	$[1,1,1,2,2,2,\frac{9}{4}]^{4}$	$[2,3,3]^{4}$	$[1,2,2]^{4}$	$[29, 22, 12, 12, 0]$	$[2, 1, 1]$	$z^{12} + 1,z^2 + 1,z + 1$	$[1, 3, 6, 12, 32]$

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