curve #688

y² + xy = x³ − 524679361811767098694165075861016437445985x + 138737907543213105519325516936534814906499021407949491249373897

a-invariants: [1, 0, 0, -524679361811767098694165075861016437445985, 138737907543213105519325516936534814906499021407949491249373897]
rank (lower bound): ≥ 25
torsion subgroup: trivial
conductor (N): 1323865339440017268000752509061793804913438776206992788931591397676827145772023933147926425771427730
naive height: 299.8044
Faltings height: 22.9247
discriminant (Δ): 928816768650172158475770114592665885679546298909840180570902743496167272199215072246600380576303985712637908929388011520000000
primes of bad reduction: 2, 3, 5, 7, 17, 31, 47, 48050387482503061, 5296867994313533401092213140427993936977679964805427501757290959300437211357
regulator: 555243174509466512352941539.4824582658131683678116333024784514629657155317781536123044118
submitted by: FNacc
submitted at: 2026-09-09 20:15:53 UTC
last updated: 2026-09-09 20:15:53 UTC

Witness: 25 independent points log in to add more points →

(-122824971751190724921174/289, 82935312826480709226375857526341979/4913)
(683599432145110962462, -9975921146584803009878225942877)
(-294368233991134514465526/361, -33870302046808625090057672852667303/6859)
(-14626705685011000307773542/17689, 6321323737860007822562845295298767775/2352637)
(-11440380738522207638294/225, -43390002306478192693246374648424699/3375)
(4038579849183496054314, -252763781921112875394282449933877)
(10272781903918567740245274/361, -32914909624766689480219820364718172103/6859)
(-93884976456593473233816/361, -110086588395171412183563392991469263/6859)
(-634991270697109758066, -14692426847651757717644762348817)
(550258323754870294727682/361, -368119160295487023438104260140707415/6859)
(3704586750416492999514/25, -1001896344115364634902158471045473/125)
(-516546327935424512739606/5041, 4951933422358238802570684066769190013/357911)
(308837946194673770929566/529, 68188980939787949501576271685849881/12167)
(493575839456744452074, -111221930901327521445973759797)
(831098637471016736474, 16635449066573600324563763966123)
(7582987289142297602237994/11881, 10348667506947053434889708071291586487/1295029)
(-693146935021406661669816/1681, 1163472251010535715253789887642130333/68921)
(2010341850800928017006271/4, 2850390880438999301587897367823008349/8)
(939888287406385840938, -21814807544445512464203517305909)
(2035246921941206420465250111/4, 91817541489467470246937263060918544045589/8)
(503944018154178262194, -1519963682041530476339587301157)
(11077589299321213902613426/9, -36869548076205907455134926617018956999/27)
(4808545694059375908802614/121, -10542629556401708072542920658109759127/1331)
(3140563188140694430447314, 5565591053044688190724768110247620123)
(26397353052942838692114/121, -7835616349365011275428933345941127/1331)

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