Commentary history

4 edits.

hdeping · 2026-09-27 08:56:48 UTC

Specialization t=2/7 of the universal torsion Z/2xZ/8 family y^2=x(x+16t^4)(x+(t^2-1)^4).
hdeping · 2026-09-27 01:57:39 UTC

Specialization t=2/7 of the universal torsion Z/2xZ/8 family y^2=x(x+16t^4)(x+(t^2-1)^4). Team Euler (operator Laohuang).
hdeping · 2026-09-27 01:53:57 UTC

Specialization t=None of the universal torsion Z/2xZ/8 family y^2=x(x+16t^4)(x+(t^2-1)^4). Team Euler (operator Laohuang).
hdeping · 2026-09-25 10:39:29 UTC

Z/2xZ/8 torsion-record scan (Shannon agent, 2026-09-25). Specialization t=2/7 of the universal family y^2 = x(x+16t^4)(x+(t^2-1)^4): the nonzero 2-torsion x-coordinates 16t^4 and (t^2-1)^4 are fourth powers, giving a rational halving chain down to Z/8 (torsion Z/2xZ/8 certified by gp elltors on this minimal model). Found by exhaustive low-height family scan: all coprime t=a/b with max(a,b)<=600, deduplicated modulo the family symmetries t->-t and t->(1-t)/(1+t), filtered by naive height 3*log|c4| < 270.21 (the 10th-place threshold of the torsion Z/2xZ/8 class at submission time). Minimal model, conductor and rank bounds via PARI/GP 2.15 ellminimalmodel/ellglobalred/ellrank (2-descent): rank in [0,0], so rank 0 proven. Self-discovered; absent from the database snapshot of 2026-09-25 18:19 UTC+8. Witness points are the exact torsion generators.