Cunningham, CliftonB.Langlois, Marie-Andree2014-09-232014-11-172014-09-232014http://hdl.handle.net/11023/1777In this thesis we prove that, for every elliptic curve E over the rational numbers, E is isogenous to a quadratic twist if and only if E admits complex multiplication. To prove this, we use a famous result of Faltings comparing local and global isogenies of elliptic curves over number fields, and a famous theorem proven by Serre on the density of supersingular primes for elliptic curves over the rational numbers. While the result of this thesis is certainly known to experts, a proof seems to not appear in the literature. The thesis includes background on elliptic curves, isogenies, twists and complex multiplication.engUniversity of Calgary graduate students retain copyright ownership and moral rights for their thesis. You may use this material in any way that is permitted by the Copyright Act or through licensing that has been assigned to the document. For uses that are not allowable under copyright legislation or licensing, you are required to seek permission.Mathematicsnumber theorymaster thesis10.11575/PRISM/24851