curve #5059

y² = x³ − 3071720608634185997709637018842244x

a-invariants: [0, 0, 0, -3071720608634185997709637018842244, 0]
c-invariants: [147442589214440927890062576904427712, 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): 49147529738146975963354192301475904
discriminant (Δ): 1854919677708369531988260618970144010828377918910916163113353941368937956098372916347518956479645618176
Faltings height: 18.3129
naive height: 242.9362
primes of bad reduction: 2, 17, 53, 71, 89, 107, 257, 263, 673
regulator: 7698895067.712369699654952226091074907501519576194987
submitted by: David Renshaw
submitted at: 2026-10-21 01:50:42 UTC
last updated: 2026-10-21 01:50:42 UTC

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

(5864898372882946609/100, 4645622717581500679939035273/1000)
(46809198933785611636532725/485276841, 262113124439040583564309666013100067285/10690163530389)
(-192202903710627563983289507/24061904161, 18295434730756403062806467049806634610665/3732458511550159)
(-2829102178332175820766261054703/1018406635050256, 94817863428382630503657152722960713114048873465/32499885446535133795904)
(-3462905263918763240068518/67782289, 2710209359398387466642295711962606280/558051585337)
(1682459725372701456023969/30250000, 182768181825209591427596211644140047/166375000000)
(-580434783190475821953899122718/16510967630161, 538978704596838618632881923274496511430946160/67090154028399672409)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 55423105368015838, 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 01:50:42 UTC · history