This requires knowing a value for the perihelion, but when you have it you can simply plug it and the semi-major axis a into the equation given. For example if the perihelion was half the distance of the semi-major axis:
e = 1 - 0.5/1 = 0.5
But if the perihelion was the same as the semi-major axis (no point on the orbit is closer than any other):
e = 1 - 1/1 = 0 (no eccentricity, perfect circle).