  Context. The 51 Eridani system harbors a complex architecture with its
primary star forming a hierarchical system with the binary GJ 3305AB at a
projected separation of 2000 au, a giant planet orbiting the primary star at 13
au, and a low-mass debris disk around the primary star with possibly a cold
component and a warm component inferred from the spectral energy distribution.
Aims. We aim to better constrain the orbital parameters of the known giant
planet. Methods. We monitored the system over three years from 2015 to 2018
with the VLT/SPHERE exoplanet imaging instrument. Results. We measure an
orbital motion for the planet of ~130 mas with a slightly decreasing separation
(~10 mas) and find a hint of curvature. This potential curvature is further
supported at 3$\sigma$ significance when including literature GPI astrometry
corrected for calibration systematics. Fits of the SPHERE and GPI data using
three complementary approaches provide broadly similar results. The data
suggest an orbital period of 32$^{+17}_{-9}$ yr (i.e. 12$^{+4}_{-2}$ au in
semi-major axis), an inclination of 133$^{+14}_{-7}$ deg, an eccentricity of
0.45$^{+0.10}_{-0.15}$, and an argument of periastron passage of
87$^{+34}_{-30}$ deg [mod 180 deg]. The time at periastron passage and the
longitude of node exhibit bimodal distributions because we do not detect yet if
the planet is accelerating or decelerating along its orbit. Given the
inclinations of the planet's orbit and of the stellar rotation axis (134-144
deg), we infer alignment or misalignment within 18 deg for the star-planet
spin-orbit. Further astrometric monitoring in the next 3-4 years is required to
confirm at a higher significance the curvature in the planet's motion,
determine if the planet is accelerating or decelerating on its orbit, and
further constrain its orbital parameters and the star-planet spin-orbit.
