$p$-adic family 2.1.8.20d1.11-1.2.8a

• Label: 2.1.8.20d1.11-1.2.8a
• Base: 2.1.8.20d1.11
• Degree: \(2\)
• e: \(2\)
• f: \(1\)
• c: \(8\)

Defining polynomial

$x^{2} + \left(b_{13} \pi^{7} + b_{11} \pi^{6} + b_{9} \pi^{5} + a_{7} \pi^{4}\right) x + c_{14} \pi^{8} + \pi$

Invariants

Residue field characteristic:	$2$
Degree:	$2$
Base field:	2.1.8.20d1.11
Ramification index $e$:	$2$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$8$
Absolute Artin slopes:	$[2,3,3,\frac{7}{2}]$
Swan slopes:	$[7]$
Means:	$\langle\frac{7}{2}\rangle$
Rams:	$(7)$
Field count:	$6$ (complete)
Ambiguity:	$2$
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:	$C_4^2:C_2$ (show 1), $C_4:D_4$ (show 2), $C_2^4.C_2^3$ (show 1), $C_2^5.(C_2\times D_4)$ (show 2)
Hidden Artin slopes:	$[\ ]^{2}$ (show 3), $[2]^{4}$ (show 1), $[2,2,3]^{4}$ (show 2)
Indices of inseparability:	$[33,26,24,8,0]$
Associated inertia:	$[1,2,1]$
Jump Set:	$[1,2,4,8,32]$

Fields

Showing all 6

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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
2.1.16.48o1.73	$x^{16} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x + 14$	$C_2^4.C_2^3$ (as 16T229)	$128$	$2$	$[2, 2, 3, 3, \frac{7}{2}]^{4}$	$[1,1,2,2,\frac{5}{2}]^{4}$	$[2]^{4}$	$[1]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 2, 4, 8, 32]$
2.1.16.48o1.74	$x^{16} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{5} + 8 x + 14$	$C_4:D_4$ (as 16T43)	$32$	$8$	$[2, 3, 3, \frac{7}{2}]^{2}$	$[1,2,2,\frac{5}{2}]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 2, 4, 8, 32]$
2.1.16.48o1.75	$x^{16} + 4 x^{12} + 4 x^{10} + 10 x^{8} + 8 x^{5} + 8 x + 14$	$C_4:D_4$ (as 16T43)	$32$	$8$	$[2, 3, 3, \frac{7}{2}]^{2}$	$[1,2,2,\frac{5}{2}]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 2, 4, 8, 32]$
2.1.16.48o1.76	$x^{16} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{7} + 8 x^{5} + 8 x + 14$	$C_4^2:C_2$ (as 16T30)	$32$	$4$	$[2, 3, 3, \frac{7}{2}]^{2}$	$[1,2,2,\frac{5}{2}]^{2}$	$[\ ]^{2}$	$[\ ]^{2}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 2, 4, 8, 32]$
2.1.16.48o1.77	$x^{16} + 4 x^{12} + 4 x^{10} + 2 x^{8} + 8 x^{3} + 8 x + 14$	$C_2^5.(C_2\times D_4)$ (as 16T833)	$512$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$	$[1,1,1,2,2,2,\frac{5}{2}]^{4}$	$[2,2,3]^{4}$	$[1,1,2]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 2, 4, 8, 32]$
2.1.16.48o1.78	$x^{16} + 4 x^{12} + 4 x^{10} + 10 x^{8} + 8 x^{3} + 8 x + 14$	$C_2^5.(C_2\times D_4)$ (as 16T833)	$512$	$2$	$[2, 2, 2, 3, 3, 3, \frac{7}{2}]^{4}$	$[1,1,1,2,2,2,\frac{5}{2}]^{4}$	$[2,2,3]^{4}$	$[1,1,2]^{4}$	$[33, 26, 24, 8, 0]$	$[1, 2, 1]$	$z^8 + 1,z^6 + 1,z + 1$	$[1, 2, 4, 8, 32]$

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