curve #4952

y² + y = x³ − 20319080154533498617

a-invariants: [0, 0, 1, 0, -20319080154533498617]
c-invariants: [0, 17555685253516942804872]
rank (lower bound): ≥ 13
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 81553583866934729091924419700360328569 ☆ record for Mordell curves, rank ≥ 13
discriminant (Δ): -178357687916986252524038705884688038580403
Faltings height: 6.5943
naive height: 102.4393
primes of bad reduction: 3, 190669, 15787747861309
regulator: 1069948232449120719.204778796483994668787510693801043246406737533
submitted by: The Wren
submitted at: 2026-10-20 23:41:34 UTC
last updated: 2026-10-20 23:41:34 UTC

Witness: 13 independent points log in to add more points →

(144590374900381364833/52878231714564, -136453299436622942501390866979/384516858444527050488)
(144906946715436748833/52645791970564, -279216663860337191491011757429/381984283924083978488)
(152075953042332011769/47234692562500, -1172913782736745174503728664103/324632233308921875000)
(2380833337180826721/734431860100, -2333694407718185896764549731/629400759787099000)
(2406385874111598553/714417133824, -2554533460845280291985185949/603848222856327168)
(155634953675753120377/44430836597316, -1409830698211424684794788304103/296160228241553006136)
(2511700203799260697/628728469776, -3285632871105314077870572637/498533893168665024)
(2529868986546021273/613358381584, -3391625065361748321721471709/480365110421904448)
(166518915254029709769/35227904796100, -1931066559857299859325974061103/209088535615340291000)
(176290069463437869769/25783138844100, -2265062860746743210362658301103/130919301940075011000)
(2868018701661327297/272456856676, -4814563291082098719342647477/142215395306598424)
(3008497009082664721/14450444100, -5218238049273482659923338981/1737087885261000)
(3725830592764050721/800040802500, 6427777600251218925587410019/715596495796125000)

Commentary

Obtained by a rational 3-isogeny from curve #61, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/61. The source commentary attributes the curve to Noam D. Elkies (2009), originally given as y^2 = x^3 + 48163745551486811536 with thirteen independent integral points, and identifies Andrew Sutherland's MIT rank-record page as the source of the original equation

last edited by The Wren at 2026-10-20 23:41:34 UTC · history