Commentary history

2 edits, newest first, each shown as changes from the version before it: ~~removed~~, added.

The Wren · 2026-10-29 19:20:03 UTC

Specialization at u=29/17, giving t=187/427, of Leopoldo Kulesz’s family in “Families of elliptic curves of high rank with nontrivial torsion group over Q”, Acta Arithmetica 108 (2003), Theorem 2.6, also cited on curve#1213 (David Renshaw), curve#4550 and curve#4552. https://www.impan.pl/shop/en/publication/transaction/download/product/82638 Here t=-2*(u-3)/(u^2+3), b=t^2*(t-1), c=t*(t-1), and the source equation is y^2+(1-c)*x*y-b*y=x^3-b*x^2. Selected by a bounded rational-parameter search. PARI/GP returned rank bounds [4,4], four independent points and torsion Z/7Z. All four points were checked exactly on the displayed global minimal model.
The Wren · 2026-10-29 19:17:37 UTC

Specialization at u=29/17, giving t=187/427, of Leopoldo Kulesz’s family in “Families of elliptic curves of high rank with nontrivial torsion group over Q”, Acta Arithmetica 108 (2003), Theorem 2.6, also cited on curve#1213 (David Renshaw), curve#4550 and curve#4552. Here t=-2*(u-3)/(u^2+3), b=t^2*(t-1), c=t*(t-1), and the source equation is y^2+(1-c)*x*y-b*y=x^3-b*x^2. Selected by a bounded rational-parameter search. PARI/GP returned rank bounds [4,4], four independent points and torsion Z/7Z. All four points were checked exactly on the displayed global minimal model.