Elliptic curves with prime conductor and a conjecture of cremona
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We find the elliptic curves defined over imaginary quadratic number fields K with class number one that have prime conductor and a K-rational 2-torsion point. Any elliptic curve with K-rational 2-torsion point has an equation of the form y2 = x3 + Ax2 + Bx. We find conditions on A and B for the elliptic curve y2 = x3 + Ax2 + Bx to have prime conductor. We also use class field theory to find primes that cannot be conductors of elliptic curves over K = Q(pd) for d = -1, -2, -3, -7 and -11.