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A classification of genus 0 modular curves with rational points, Ph.D. Thesis 

Abstract : Let E be a non-CM elliptic curve defined over Q. Fix an algebraic closure Q¯ of Q. We get a Galois representation ρ_E:Gal(Q¯/Q)→GL(2,Z^) associated to E by choosing a compatible bases for the N-torsion subgroups of E(Q¯). Associated to an open subgroup G of GL(2,Z^) such that G contains -I and has full determinant, we have the modular curve (X_G,π_G) over Q which loosely parametrises elliptic curves E such that image of ρ_E is conjugate to a subgroup of G^t. In this article we give a complete classification of all such genus 0 modular curves that have a rational point. This classification is given in finitely many families. Moreover, each such modular curve can be explicitly computed.

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