On construction of certain CM elliptic curves

Abstract

Let E be a CM elliptic curve defined over an algebraic number field F. In the previous paper [N. Murabayashi, On the field of definition for modularity of CM elliptic curves, J. Number Theory 108 (2004) 268–286], we gave necessary and sufficient conditions for E to be modular over F, i.e. there exists a normalized newform f of weight two on Γ 1 ( N ) for some N such that Hom F ( E , J f ) ≠ { 0 } . We also determined the multiplicity of E as F-simple factor of J f when Hom F ( E , J f ) ≠ { 0 } . In this process we separated into the three cases. In this paper we construct certain CM elliptic curves which satisfy the conditions of each case. In other words, we show that all three cases certainly occur.