**Andrej Dujella**

Univ. of Zagreb, Croatia

January 17, 2013

**Elliptic curves with large rank**:
There are exactly 15 possible torsion groups for elliptic curves over Q, but little is known about which values of rank are possible. We will describe some of the methods used in construction of elliptic curves with relatively high rank and prescribed torsion group. We will also present some recent results concerning elliptic curves over quadratic and quartic fields. We will discuss possible applications of elliptic curves with large torsion and positive rank over number fields of small degree (instead over Q) in Elliptic curve factorization method (ECM).