curve #448

y² + xy = x³ − 74494006217084732231494701201553460x + 7786959773639428479142830515805837724808339207958672

a-invariants: [1, 0, 0, -74494006217084732231494701201553460, 7786959773639428479142830515805837724808339207958672]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 5597199175118491536130878384121905070868151610189056358131211447741842709470
discriminant (Δ): 262132484055168836760776897065169957848979652219168298752992435420665593103233127060118979791769600000000
Faltings height: 18.8774
naive height: 252.5017
primes of bad reduction: 2, 3, 5, 7, 13, 17, 29, 79, 293, 4969, 577873, 1112129, 2081801, 161195362687, 127704680002349, 1312838632356181391
regulator: 80585923170408260509.4439505519871773282866932461825816391224276517472502299023
submitted by: Bhavik Mehta
submitted at: 2026-08-30 15:55:01 UTC
last updated: 2026-08-30 15:55:01 UTC

Witness: 20 independent points log in to add more points →

(143328833763702184, 7365264967233225868986508)
(211681681290884584, 38771331505503406340173708)
(-86849014521118616, 116625941763599983922068108)
(657367978305645256/9, 1412160022537238039002482116/27)
(167340752077110664, -2666020289432732823067412)
(166859614356416488, 1626606739477569271567756)
(146786267817812584, -3866541915392189845714292)
(201469143426337384, -30924287679678360054043892)
(288825516815339584, 101808969441344612700406708)
(147038410111124584, -3534572130368119203538292)
(141549522230976184, -8859618583682490364661492)
(-232174088258307416, 112103694389737149237781708)
(115652613551838184, -26803878850670191783685492)
(148249760095449064, -1213742693743867010801012)
(3672680386953849256/9, 5749342101630028560377298116/27)
(148336470899111638, 856894609833524368413736)
(77117944020970552936/529, -61205693163657289604259710764/12167)
(204702475690244584, -33399420957627968352466292)
(169353449692868584, 5319760375436060031757708)
(13678960832728296664/121, -37886821218558886232409012652/1331)

Commentary

Found by specializing Noam D. Elkies's rank-17 elliptic K3 fibration (arXiv:2608.25406) at t = 5/16. The 3 points beyond the seventeen generic sections came from searching the 1311 norm-4 quartic covers of the Mordell-Weil lattice; the claimed lower bound is rank >= 20.

last edited by Bhavik Mehta at 2026-08-30 15:55:01 UTC · history