curve #418

y² + xy = x³ − 815073141784200754215404771679519674339839601501320x + 10148734322864283657618226218200352363193275261694417405081164533181261838400

a-invariants: [1, 0, 0, -815073141784200754215404771679519674339839601501320, 10148734322864283657618226218200352363193275261694417405081164533181261838400]
rank (lower bound): ≥ 21
torsion subgroup: trivial
conductor (N): 78215928850816814315475228669912320288965457568304335441059486388743780155523303370660512822073067247701057863810237592566930
discriminant (Δ): -9839316513169517755920678269709069435748137552838716848014248656337848459529179209312843801536495603592215241976105162110825387465187436111452309913600000
Faltings height: 28.2759
naive height: 363.5456
primes of bad reduction: 2, 3, 5, 7, 13, 19, 41, 43, 97, 100291, 12035614027, 136275301181889479, 53605277496717250253987332261938249194470530456940125692776944124494415356042639083
regulator: 110433344918437932892855708.09282462382429890159132163575510608127887573793464860
submitted by: 7fff-zip
submitted at: 2026-08-29 12:46:00 UTC
last updated: 2026-08-29 12:46:00 UTC

Witness: 21 independent points log in to add more points →

(-24193508905300590986569520, 125328077111701337398503316998977565880)
(7316967155555376927038840, -67650622789190337967249127905264805320)
(3030145243092182573283672928/9, 166221156943769535222386584652895806113592/27)
(-8058481787692539451029488, -127254373620120059944647124767302342024)
(-133356623944165889347670240/9, 3719024356874097977774728808420741459960/27)
(-27436290212495229969200560, 108897426792187636691528695100525255480)
(22142803730800206405428240, 54382254266982487233889894163537615480)
(-21847915482072618180692440, -132392171648216317435693360703937850720)
(22553058957892040455821206600, -3386942588678979413347388688249237671751880)
(1249624670206398066102169840, 44162855241023436099699754654966659158680)
(263016412784901712238823440, -4241535582865718771892073686401684304520)
(18414303912818040397247360, 37199103134317927234953814129256330360)
(519845112470884862648459210, -11835067790366097054330785169242821864990)
(56398976758383860961014435/4, -305625125424316624299280857866033500085/8)
(-11109213354771312142122310, 133538445381408206562089552031786490580)
(17533735018800278430757940, -35325439395682660875327399924999823420)
(93836663343319906442405840, -871737793894317089847017629827291048520)
(5953436900739983749707097640, -459353097505040201928408381215711492103520)
(25925459376412850961181940, -80267386214363440535026830312203324920)
(446770149468456587157768422373129560/14641242001, -207287550354291895707005818896537906916774914194915720/1771604923363001)
(182709864693850184548404331808/5929, -54585252132977821641470049096720287073551912/456533)

Commentary

Found in this search on Noam D. Elkies's published q1/q2 rank-19 pullback at t=-5440/1843 (u=849006/1843, v=10523171/1843).

last edited by 7fff-zip at 2026-08-29 17:30:30 UTC · history