curve #5055

y² = x³ − 290935282793986863680753548423561x

a-invariants: [0, 0, 0, -290935282793986863680753548423561, 0]
c-invariants: [13964893574111369456676170324330928, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 9309929049407579637784113549553952
discriminant (Δ): 1576046956989585601402663805276104231194425908163285389159718332625619225881794131482554136864478784
Faltings height: 17.7237
naive height: 235.8656
primes of bad reduction: 2, 7, 31, 67, 107, 977, 1487, 7547
regulator: 45396935457.19635715968066623565060717986217236807722
submitted by: David Renshaw
submitted at: 2026-10-21 01:50:27 UTC
last updated: 2026-10-21 01:50:27 UTC

Witness: 7 independent points log in to add more points →

(-5307214321900018534412367/3639950224, 142506413026894970076467631712186182807/219605476914368)
(-2457150790226314929152/214369, 134223698085495621930146235325728/99252847)
(-207915001759990543856950336/250063003969, 61429073079322739399977689380651752382040/125047255953750047)
(-838380493904195385470439871/255861965584, 124010734790998078747960243268385518446305/129422146327423552)
(3039532731775075485306619904187392/132833451124518169, 111705644594208864931020191512731320605704067293216/48412909578723181939053053)
(1499302944197683110138841393493761/65355740379745344, 38821691171364667903624042652035908109491124155775/16708043925610139222364672)
(-1765100583680914200538734594940638992/11995888183212866862609, 271831119947472753336831580057326830191946871176089175828/1313858555450997669945388067726727)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 17056825108852669, rank 7 example from Watkins, Donnelly, Elkies, Fisher, Granville, Rogers, 'Ranks of quadratic twists of elliptic curves', Publ. Math. Besancon 2014, doi:10.5802/pmb.9; points from https://magma.maths.usyd.edu.au/~watkins/PTS.r6r7.

last edited by David Renshaw at 2026-10-21 01:50:27 UTC · history