This question was underspecified, so I'm going to try again.
Osculating orbital elements are Keplerian elements of a hypothetical orbit around a specified center which would be tangent to a specified body's location and direction of motion.
Below I've used Horizons to generate osculating elements for Mercury around the solar system barycenter and around the Sun. Since gravity from all planets contribute to Mercury's motion both directly and by their effect on the movement of Sun and each other, neither is the correct orbit of Mercury.
- Is it possible to determine the mass of the body that would be at each center to produce these orbital elements?
- Is it possible without using the period included in the output?
Output for Mercury from JPL's Horizons
For 2458849.500000000 = A.D. 2020-Jan-01 00:00:00.0000 TDB Keplerian osculating element barycenter (@0) Sun (@sun) Eccentricity EC: 2.200014479776101E-01 2.056502791763153E-01 Inclination wrt XY-plane IN: 7.072029216276680E+00 7.003793158072573E+00 Long. of Ascend. Nd, OMEGA OM: 4.754470515340162E+01 4.830652204584498E+01 Argument of Perifocus, w W: 2.767991909564741E+01 2.918253289128074E+01 Mean anomaly, M MA: 1.899912221742112E+02 1.872506756791482E+02 True anomaly, nu TA: 1.865562987479777E+02 1.848845489756656E+02 Semi-major axis, a (AU) A: 3.772942492110727E-01 3.870976616195002E-01 Sidereal orbit period (day) PR: 8.448193239598780E+01 8.796891036601312E+01