curve #5053

y² = x³ − 58010900619086265938339810545156x

a-invariants: [0, 0, 0, -58010900619086265938339810545156, 0]
c-invariants: [2784523229716140765040310906167488, 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): 928174409905380255013436968722496
discriminant (Δ): 12494209902361742735086819498697817049767824272563377793991421041957942818634923696494142282010624
Faltings height: 17.3206
naive height: 231.0282
primes of bad reduction: 2, 13, 53, 61, 1049, 2203, 39209
regulator: 26469833199.00466735590707397429714375638253943540592
submitted by: David Renshaw
submitted at: 2026-10-21 01:50:20 UTC
last updated: 2026-10-21 01:50:20 UTC

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

(-6630781716224644296401654/2119405369, 37898524807046727305935501089494234040/97571064972653)
(-47959503705388406/25, 40355252772076974950450328/125)
(-2012369763426013460703016016078/408976726187929, 3372936566409678481200200962922513596582189344/8270807086672118678717)
(14367866432533013088026466453/1642849737169, 846450798696590930283655136383738791340827/2105701293569782553)
(117995513044227133795416079244401/450027177256336, 1281194088660299280511666189048097856608790452745/9546806334076259915584)
(113068860380488414690384383157/7063704164169, 33440362264229828518927516458960880604489141/18773651570474293947)
(11050833054617576728085564813607361/216241706842705296, 1148721515899435589086781050415332173004781411416415/100556278247879684997937344)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 7616488732945534, 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 01:50:20 UTC · history