$p$-adic family 2.1.2.2a1.1-1.4.18a

• Label: 2.1.2.2a1.1-1.4.18a
• Base: 2.1.2.2a1.1
• Degree: \(4\)
• e: \(4\)
• f: \(1\)
• c: \(18\)

Defining polynomial

$x^{4} + \left(b_{19} \pi^{5} + a_{15} \pi^{4}\right) x^{3} + \left(c_{22} \pi^{6} + b_{14} \pi^{4} + b_{10} \pi^{3}\right) x^{2} + \left(b_{21} \pi^{6} + b_{17} \pi^{5}\right) x + c_{16} \pi^{5} + \pi$

Invariants

Residue field characteristic:	$2$
Degree:	$4$
Base field:	$\Q_{2}(\sqrt{-1})$
Ramification index $e$:	$4$
Residue field degree $f$:	$1$
Discriminant exponent $c$:	$18$
Absolute Artin slopes:	$[2,\frac{7}{2},\frac{17}{4}]$
Swan slopes:	$[4,\frac{11}{2}]$
Means:	$\langle2,\frac{15}{4}\rangle$
Rams:	$(4,7)$
Field count:	$32$ (complete)
Ambiguity:	$4$
Mass:	$32$
Absolute Mass:	$16$

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 16), $C_2 \wr C_2\wr C_2$ (show 16)
Hidden Artin slopes:	$[3,4]$ (show 16), $[2,3,4]^{2}$ (show 16)
Indices of inseparability:	$[19,12,4,0]$
Associated inertia:	$[1,1,1]$
Jump Set:	$[1,5,13,21]$

Fields

Showing all 16

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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
2.1.8.26b1.1	$x^{8} + 2 x^{4} + 8 x^{3} + 2$	$C_2 \wr C_2\wr C_2$ (as 8T35)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,3,4]^{2}$	$[1,2,3]^{2}$	$[19, 12, 4, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 5, 13, 21]$
2.1.8.26b1.2	$x^{8} + 2 x^{4} + 8 x^{3} + 16 x^{2} + 2$	$C_2 \wr C_2\wr C_2$ (as 8T35)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,3,4]^{2}$	$[1,2,3]^{2}$	$[19, 12, 4, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 5, 13, 21]$
2.1.8.26b1.3	$x^{8} + 2 x^{4} + 8 x^{3} + 16 x + 2$	$C_2 \wr C_2\wr C_2$ (as 8T35)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,3,4]^{2}$	$[1,2,3]^{2}$	$[19, 12, 4, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 5, 13, 21]$
2.1.8.26b1.4	$x^{8} + 2 x^{4} + 8 x^{3} + 16 x^{2} + 16 x + 2$	$C_2 \wr C_2\wr C_2$ (as 8T35)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,3,4]^{2}$	$[1,2,3]^{2}$	$[19, 12, 4, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 5, 13, 21]$
2.1.8.26b1.9	$x^{8} + 10 x^{4} + 8 x^{3} + 2$	$C_2 \wr C_2\wr C_2$ (as 8T35)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,3,4]^{2}$	$[1,2,3]^{2}$	$[19, 12, 4, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 5, 13, 21]$
2.1.8.26b1.10	$x^{8} + 10 x^{4} + 8 x^{3} + 16 x^{2} + 2$	$C_2 \wr C_2\wr C_2$ (as 8T35)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,3,4]^{2}$	$[1,2,3]^{2}$	$[19, 12, 4, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 5, 13, 21]$
2.1.8.26b1.15	$x^{8} + 10 x^{4} + 8 x^{3} + 8 x^{2} + 2$	$C_2 \wr C_2\wr C_2$ (as 8T35)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,3,4]^{2}$	$[1,2,3]^{2}$	$[19, 12, 4, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 5, 13, 21]$
2.1.8.26b1.16	$x^{8} + 8 x^{7} + 10 x^{4} + 8 x^{3} + 8 x^{2} + 2$	$C_2 \wr C_2\wr C_2$ (as 8T35)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,3,4]^{2}$	$[1,2,3]^{2}$	$[19, 12, 4, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 5, 13, 21]$
2.1.8.26b1.17	$x^{8} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 2$	$C_2 \wr C_2\wr C_2$ (as 8T35)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,3,4]^{2}$	$[1,2,3]^{2}$	$[19, 12, 4, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 5, 13, 21]$
2.1.8.26b1.18	$x^{8} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 16 x^{2} + 2$	$C_2 \wr C_2\wr C_2$ (as 8T35)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,3,4]^{2}$	$[1,2,3]^{2}$	$[19, 12, 4, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 5, 13, 21]$
2.1.8.26b1.23	$x^{8} + 4 x^{6} + 10 x^{4} + 8 x^{3} + 2$	$C_2 \wr C_2\wr C_2$ (as 8T35)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,3,4]^{2}$	$[1,2,3]^{2}$	$[19, 12, 4, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 5, 13, 21]$
2.1.8.26b1.24	$x^{8} + 4 x^{6} + 10 x^{4} + 8 x^{3} + 16 x^{2} + 2$	$C_2 \wr C_2\wr C_2$ (as 8T35)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,3,4]^{2}$	$[1,2,3]^{2}$	$[19, 12, 4, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 5, 13, 21]$
2.1.8.26b1.25	$x^{8} + 4 x^{6} + 10 x^{4} + 8 x^{3} + 16 x + 2$	$C_2 \wr C_2\wr C_2$ (as 8T35)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,3,4]^{2}$	$[1,2,3]^{2}$	$[19, 12, 4, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 5, 13, 21]$
2.1.8.26b1.26	$x^{8} + 4 x^{6} + 10 x^{4} + 8 x^{3} + 16 x^{2} + 16 x + 2$	$C_2 \wr C_2\wr C_2$ (as 8T35)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,3,4]^{2}$	$[1,2,3]^{2}$	$[19, 12, 4, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 5, 13, 21]$
2.1.8.26b1.31	$x^{8} + 4 x^{6} + 10 x^{4} + 8 x^{3} + 8 x^{2} + 2$	$C_2 \wr C_2\wr C_2$ (as 8T35)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,3,4]^{2}$	$[1,2,3]^{2}$	$[19, 12, 4, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 5, 13, 21]$
2.1.8.26b1.32	$x^{8} + 8 x^{7} + 4 x^{6} + 10 x^{4} + 8 x^{3} + 8 x^{2} + 2$	$C_2 \wr C_2\wr C_2$ (as 8T35)	$128$	$2$	$[2, 2, 3, \frac{7}{2}, 4, \frac{17}{4}]^{2}$	$[1,1,2,\frac{5}{2},3,\frac{13}{4}]^{2}$	$[2,3,4]^{2}$	$[1,2,3]^{2}$	$[19, 12, 4, 0]$	$[1, 1, 1]$	$z^4 + 1,z^2 + 1,z + 1$	$[1, 5, 13, 21]$

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