The aim of this thesis is to de�ne the Elliptic Curves and some of intresting properties of a special
class of terms,namely,rational point of elliptic curve ,The properties of the rational pooint of curve
is airthmetic ,and rational point on elliptic curve forms a �nitely generated group structure group it
will be done by using the chord-tangent group law of composition.After completing the the abelian
group structure,so we look some elementary properties like for a given elliptic curve we �nd the
torsion subgroup of that elliptic curves.and �nally done the elliptic curve and their isomorphism.
Further, we study different families of elliptic curves which depend on different parameters.
Moreover, we look at the reduction modulo p of an elliptic curve and infer the meaning of good and
bad reduction of an elliptic curve. Lastly, the statement and proof of Mordell-Weil theorem is given.
The topics of the thesis are based on the book Elliptic Curves by Dale Husemoller. I have
not added anything new, except making a few observations of my own
