E has torsion Z/3Z and rank exactly 7. It is isomorphic to the Tate form y^2 + 849xy - 3834921y = x^3 (a1 = 849, a3 = -3834921), in which (0,0) is a point of order 3.
Every curve with a rational point of order 3 has such a form y^2 + a1 xy + a3 y = x^3, and (a1, a3) and (a1/u, a3/u^3) give isomorphic curves (x -> u^2 x, y -> u^3 y), so the curve is determined by t = a3/a1^3: it is the member t = -1278307/203986683 (in lowest terms) of the one-parameter Z/3 family y^2 + xy + ty = x^3. The submitted a-invariants are the global minimal model of this curve (the Tate form above is already minimal; the two differ by a translation), on which the point (0,0) of the Tate form is (60067, -1947494).
It was found by a screen of the discriminant cusps of the Z/3 family: all minimal models y^2 + a1 xy + a3 y = x^3 with |Delta| = |a3|^3 |a1^3 - 27 a3| <= 1.18 x 10^30 (ten times the previous T[3] rank-7 discriminant record). Most of these have a3 small against a1^3 (|t| small) and lie outside the boxes searched before. A Mestre-Nagao screen in C over 3.6 x 10^11 models left 1,745 new candidates for Magma (2-Selmer rank, 2-descent with point search, RankBounds). This is the discriminant-first search of the integral Tate X_1(3) family that jesper-petersen used at rank 3 (#3759, https://elliptic-rank.icarm.cloud/curve/3759), carried to rank 7.
Search scoped and carried out by Claude code Opus 5.5
Lewis Combes ·