$p$-adic family 2.1.4.11a1.12-1.4.12b

• Label: 2.1.4.11a1.12-1.4.12b
• Base: 2.1.4.11a1.12
• 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:	2.1.4.11a1.12
Ramification index $e$:	$4$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$12$
Absolute Artin slopes:	$[2,3,4,4]$
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:	$C_2^4.C_2^3$ (show 2), $C_2^4.D_4$ (show 2)
Hidden Artin slopes:	$[3,\frac{7}{2}]^{2}$
Indices of inseparability:	$[41,36,20,8,0]$
Associated inertia:	$[1,1,2]$
Jump Set:	$[1,2,4,8,32]$ (show 2), $[1,9,32,48,64]$ (show 2)

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
2.1.16.56l1.2	$x^{16} + 8 x^{9} + 2 x^{8} + 4 x^{4} + 18$	$C_2^4.C_2^3$ (as 16T231)	$128$	$2$	$[2, 3, 3, \frac{7}{2}, 4, 4]^{2}$	$[1,2,2,\frac{5}{2},3,3]^{2}$	$[3,\frac{7}{2}]^{2}$	$[2,\frac{5}{2}]^{2}$	$[41, 36, 20, 8, 0]$	$[1, 1, 2]$	$z^8 + 1,z^4 + 1,z^3 + 1$	$[1, 9, 32, 48, 64]$
2.1.16.56l1.6	$x^{16} + 8 x^{15} + 8 x^{14} + 8 x^{9} + 2 x^{8} + 4 x^{4} + 18$	$C_2^4.C_2^3$ (as 16T231)	$128$	$2$	$[2, 3, 3, \frac{7}{2}, 4, 4]^{2}$	$[1,2,2,\frac{5}{2},3,3]^{2}$	$[3,\frac{7}{2}]^{2}$	$[2,\frac{5}{2}]^{2}$	$[41, 36, 20, 8, 0]$	$[1, 1, 2]$	$z^8 + 1,z^4 + 1,z^3 + 1$	$[1, 9, 32, 48, 64]$
2.1.16.56l1.236	$x^{16} + 8 x^{15} + 4 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 4 x^{4} + 14$	$C_2^4.D_4$ (as 16T268)	$128$	$2$	$[2, 3, 3, \frac{7}{2}, 4, 4]^{2}$	$[1,2,2,\frac{5}{2},3,3]^{2}$	$[3,\frac{7}{2}]^{2}$	$[2,\frac{5}{2}]^{2}$	$[41, 36, 20, 8, 0]$	$[1, 1, 2]$	$z^8 + 1,z^4 + 1,z^3 + 1$	$[1, 2, 4, 8, 32]$
2.1.16.56l1.238	$x^{16} + 8 x^{14} + 4 x^{12} + 8 x^{11} + 8 x^{9} + 2 x^{8} + 4 x^{4} + 14$	$C_2^4.D_4$ (as 16T268)	$128$	$2$	$[2, 3, 3, \frac{7}{2}, 4, 4]^{2}$	$[1,2,2,\frac{5}{2},3,3]^{2}$	$[3,\frac{7}{2}]^{2}$	$[2,\frac{5}{2}]^{2}$	$[41, 36, 20, 8, 0]$	$[1, 1, 2]$	$z^8 + 1,z^4 + 1,z^3 + 1$	$[1, 2, 4, 8, 32]$

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