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We develop an algorithm for bounding the rank of elliptic curves in the family y2 = x3 -Bx, all of them with torsion group Z/(2Z) and modular invariant j = 1728. We use it to look for curves of high rank in this family and present four such curves of rank 13 and 22 of rank 12.

High rank elliptic curves with torsion group $\mathbb {Z}/(2\mathbb {Z})$