In this work, we derive some properties of n-universal quadratic forms, quadratic ideals and elliptic curves over finite fields F(p) for primes p >= 5. In the first section, we give sonic preliminaries form binary quadratic forms and quadratic idelas. In the second section, we consider the quadratic ideals and quadratic forms. In the third section, we consider the quadratic forms over finite fields, also consider the representations of positive integers by quadratic forms and n-universal forms. In the last section, we consider the number of rational points on elliptic curves associated with the universal forms.