$p$-adic family 2.1.2.2a1.1-2.4.12a

• Label: 2.1.2.2a1.1-2.4.12a
• Base: 2.1.2.2a1.1
• Degree: \(8\)
• e: \(4\)
• f: \(2\)
• c: \(12\)

Defining polynomial over unramified subextension

$x^{4} + a_{3} \pi x^{3} + b_{2} \pi x^{2} + c_{4} \pi^{2} + \pi$

Invariants

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

Diagrams

Varying

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

Galois group:	$C_2^3:C_4$ (show 2), $C_2^2\times A_4$ (show 1), $C_2\wr C_4$ (show 2), $C_2^4:A_4$ (show 2), $C_2^5:C_8$ (show 2)
Hidden Artin slopes:	$[2,2]^{4}$ (show 2), $[2,2]^{3}$ (show 2), $[\ ]^{3}$ (show 1), $[\ ]^{2}$ (show 2), $[2]^{2}$ (show 2)
Indices of inseparability:	$[7,6,4,0]$ (show 6), $[7,6,6,0]$ (show 1), $[7,7,4,0]$ (show 1), $[7,7,7,0]$ (show 1)
Associated inertia:	$[2]$ (show 4), $[3]$ (show 3), $[4]$ (show 2)
Jump Set:	$[1,2,7,14]$ (show 3), $[1,3,7,14]$ (show 6)

Fields

Showing all 9

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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
2.2.8.28a1.1	$( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2$	$C_2^2\times A_4$ (as 16T58)	$48$	$4$	$[2, 2, 2]^{6}$	$[1,1,1]^{6}$	$[\ ]^{3}$	$[\ ]^{3}$	$[7, 7, 7, 0]$	$[3]$	$z^7 + 1$	$[1, 3, 7, 14]$
2.2.8.28a6.1	$( x^{2} + x + 1 )^{8} + \left(2 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2$	$C_2^5:C_8$ (as 16T587)	$256$	$2$	$[2, 2, 2, 2, 2]^{8}$	$[1,1,1,1,1]^{8}$	$[2,2]^{4}$	$[1,1]^{4}$	$[7, 6, 6, 0]$	$[4]$	$z^7 + t z + t$	$[1, 3, 7, 14]$
2.2.8.28a10.1	$( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 2$	$C_2^3:C_4$ (as 16T33)	$32$	$8$	$[2, 2, 2]^{4}$	$[1,1,1]^{4}$	$[\ ]^{2}$	$[\ ]^{2}$	$[7, 6, 4, 0]$	$[2]$	$z^7 + z^3 + z + 1$	$[1, 2, 7, 14]$
2.2.8.28a10.3	$( x^{2} + x + 1 )^{8} + 2 ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 6$	$C_2^3:C_4$ (as 16T33)	$32$	$8$	$[2, 2, 2]^{4}$	$[1,1,1]^{4}$	$[\ ]^{2}$	$[\ ]^{2}$	$[7, 6, 4, 0]$	$[2]$	$z^7 + z^3 + z + 1$	$[1, 2, 7, 14]$
2.2.8.28a14.1	$( x^{2} + x + 1 )^{8} + 2 x ( x^{2} + x + 1 )^{7} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{4} + 2$	$C_2^4:A_4$ (as 16T416)	$192$	$2$	$[2, 2, 2, 2, 2]^{6}$	$[1,1,1,1,1]^{6}$	$[2,2]^{3}$	$[1,1]^{3}$	$[7, 6, 4, 0]$	$[3]$	$z^7 + z^3 + t z + (t + 1)$	$[1, 2, 7, 14]$
2.2.8.28a17.1	$( x^{2} + x + 1 )^{8} + \left(2 x + 2\right) ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 2$	$C_2^5:C_8$ (as 16T587)	$256$	$2$	$[2, 2, 2, 2, 2]^{8}$	$[1,1,1,1,1]^{8}$	$[2,2]^{4}$	$[1,1]^{4}$	$[7, 7, 4, 0]$	$[4]$	$z^7 + t z^3 + (t + 1)$	$[1, 3, 7, 14]$
2.2.8.28a19.1	$( x^{2} + x + 1 )^{8} + 2 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 2$	$C_2\wr C_4$ (as 16T171)	$64$	$4$	$[2, 2, 2, 2]^{4}$	$[1,1,1,1]^{4}$	$[2]^{2}$	$[1]^{2}$	$[7, 6, 4, 0]$	$[2]$	$z^7 + t z^3 + z + t$	$[1, 3, 7, 14]$
2.2.8.28a19.3	$( x^{2} + x + 1 )^{8} + 2 x ( x^{2} + x + 1 )^{7} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 6$	$C_2\wr C_4$ (as 16T171)	$64$	$4$	$[2, 2, 2, 2]^{4}$	$[1,1,1,1]^{4}$	$[2]^{2}$	$[1]^{2}$	$[7, 6, 4, 0]$	$[2]$	$z^7 + t z^3 + z + t$	$[1, 3, 7, 14]$
2.2.8.28a24.1	$( x^{2} + x + 1 )^{8} + 2 x ( x^{2} + x + 1 )^{6} + 2 ( x^{2} + x + 1 )^{5} + 2 x ( x^{2} + x + 1 )^{4} + 2$	$C_2^4:A_4$ (as 16T416)	$192$	$2$	$[2, 2, 2, 2, 2]^{6}$	$[1,1,1,1,1]^{6}$	$[2,2]^{3}$	$[1,1]^{3}$	$[7, 6, 4, 0]$	$[3]$	$z^7 + t z^3 + (t + 1) z + 1$	$[1, 3, 7, 14]$

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