  We confirm the trend in the radial velocity data for Epsilon Indi A
suggesting a long-period planetary companion and find significant curvature is
present, sufficient to quantify Epsilon Indi Ab as a cold Jupiter with a
minimum mass of $2.71_{-0.44}^{+2.19}~M_{\rm Jup}$ on a nearly circular orbit
with a semi-major axis of $12.82_{-0.71}^{+4.18}$ au and an orbital period of
$52.62_{-4.12}^{+27.70}$ yr. We also identify other significant signals in the
radial velocity data. We investigate a variety of spectral diagnostics and
interpret these signals as arising from activity-induced radial velocity
variations. In particular, the 2500 and 278 d signals are caused by magnetic
cycles. While a planetary signal might be present in the 17.8 d signal, the
origin of 17.8 and 11 d signals are most easily interpreted as arising in the
rotation of the star with a period of about 35 d. We find that traditional
activity indicators have a variety of sensitivities. In particular, the sodium
lines and CaHK index are sensitive to all activity-induced signals. The line
bisector measurement is sensitive to stellar rotation signal while H$\alpha$ is
sensitive to the secondary magnetic cycle. In general, because of their
different sensitivities these activity indicators introduce extra noise if
included in the noise model whereas differential RVs provide a robust proxy to
remove wavelength-dependent noise efficiently. Based on these analyses, we
propose an activity diagnostics procedure for the detection of low amplitude
signals in high precision radial velocity data. Thus the Epsilon Indi system
comprises of at least Epsilon Indi A, Ab as well as a long period brown dwarf
binary Ba and Bb; so it provides a benchmark case for our understanding of the
formation of gas giants and brown dwarfs.
