Rank 13, torsion Z/2Z. Fibre (a,b,c,u) = (45,130,170,157) of Kihara's Z/2Z construction (six points from the quartic y^2 = A t^4 + B t^2 + C at t = u±a, u±b, u±c), shown as its minimal model. Found by an exhaustive search of every primitive fibre with a<b<c<=200 and u<=200 (about 210 million fibres), ranked by a Nagao-type sum over p<1000. Then the exact minimal-model size was computed and PARI's ellrank was run. The 2-descent upper bound is 13, and the 13 listed points are independent (saturated at primes up to 200, LLL-reduced), so the rank is exactly 13. Minimal discriminant: log|Δ| = 137.627, compared with 140.786 for curve#2158.
Commentary
last edited by Steps Unbounded at · history
Log in to edit commentary.