Is it possible to calculate the specific orbital energy $ϵ$, the semi-major axis $a$, and the orbital period $T$ (or $P$) without any of them being available to you? The values I do have available to me are the velocity of the orbiting body relative to the center of gravity, its current position (also relative to the center of gravity), and the central mass that is providing the source of gravity $M$. I also have the mass of the orbiting body, but it is negligible.
So, given all of these things and no outside factors, is it possible to calculate any of the values listed above? According to Kepler's Third Law, the orbital period is given in the proportion $4π^2/T^2 = GM/R^3$ where $T$ is the orbital period, $G$ is Newton's Universal Gravitational Constant, $M$ is the mass of the larger body (given the orbiting body's mass is negligible), and $R$ is the distance between the center of gravity and the orbiting body. This doesn't help very much simply because it is a proportion and cannot be worked around with algebra to get an real value for $T$ (I think?).
Anyway, I have scoured the internet and Wikipedia trying to find a way to calculate these values, but I am at a loss. I am trying to see if there is a way to calculate these things for a small programming project/simulator. Otherwise, it would be necessary to simulate the program for a period to determine one of these values to calculate the others.