Elliptic Curve Rank Leaderboard

Found by a conductor-oriented search based on Thomas J. Kretschmer’s Construction of Elliptic Curves with Large Rank (Math. Comp. 46 (1986), 627–635). I searched curves of the form \(y^2=x^3+ax^2+bx\) with rational 2-torsion, taking \(b=2^{e_2}3^{e_3}pqr\) and looking for values of \(a\) for which enough of Kretschmer’s conditions \(b_1+a+b_2=\square\) are satisfied to force rank at least 5. Candidates were then tested for exact rank and conductor. This produced \(y^2=x^3+1637x^2+538080x\), whose minimal model has ainvs \([1,1,0,-22198,480352]\), exact rank 5, torsion \(\mathbf Z/2\mathbf Z\), and conductor \(1364469990\), independently confirmed in Magma.