curve #629

y² + xy = x³ − 206146472796129715074018059573733454563478276563335x + 1105311419338647732650534549181813400819706534279793673221000968292289116297

a-invariants

[1, 0, 0, -206146472796129715074018059573733454563478276563335, 1105311419338647732650534549181813400819706534279793673221000968292289116297]

rank (lower bound)

≥ 27

torsion subgroup

trivial

conductor (N)

1501492496847230651915535798135038791367857964464751861873735344250891303798770296273960611057947540201444218392271650661570

discriminant (Δ)

32890330752280708732651287188959919374134379111932748195500602422159240389543744044056588696059474384051705456392854745943699010222081672024268544000000

Faltings height

27.8456

naive height

359.1716

primes of bad reduction

2, 3, 5, 7, 11, 13, 17, 41, 43, 59, 20615041675346507, 1371609143868959976154145640664687027995574838295626697379856471001437756163360865284294725678053

regulator

35972256687617507862791281178743.974666367850133064664882815135281207363181440640813284688707

submitted by

Roy van Rijn

submitted at

2026-09-07 08:27:41 UTC

later contributions

primes of bad reduction recorded · Roy van Rijn · 2026-09-07 08:41:48 UTC

Witness: 27 independent points log in to add more points →

(26321682467069393430444334, 117964681837542586078930769286934810333)
(54887205997117718194309684, 393883231047691220483033416848114209383)
(32364588736345584752030914, 168327928353290521161961331587947907873)
(837382870832642084345680888/49, -17399741452579785454071230206148202234611/343)
(-804270223134201944511558422/49, -2806392353782498054373666055715045522325/343)
(3977895807020400579531034, -18660843513949020619038104535278008967)
(-26863657709572974942723161510426/6255001, 683945839477548738377797550837592311808123164273/15643757501)
(13951773433085292719372374, -30739923341158156237811266635921322427)
(113758770705775043939355751894/5041, 31903306954368715456595802988845236871578963/357911)
(-4288200704829464117927643338263721508404/821965058224489, -1064059717130686371704105468915519590402071426672870934545749/23565683147637198589237)
(-2178079240602961904294398989887546/247779081, 184520308304141032332941691393326940127937789907193/3900290514021)
(151413054202398477204441166/25, 1110622132112706025604182302788425532921/125)
(74572808369071851609647254, 632802903068079169003727767441564930693)
(22244725452057537034310896126290166/3186263809, -454674016518409386852332853894707865249750897385381/179855033226623)
(95377918106874621452179114600580086/70946929, -29454190170646907786725031005785788506159832897448589/597585982967)
(16669331397757938290437711/4, 142795656402141019327524237442251526289/8)
(522424435058999243556508024/1225, -1367621047337013272213148664462435785663463/42875)
(521832741619123424673557253916034806/99422287969, -406225249806291108637986977157117913922158425787828619/31349139886369297)
(91164493608561500077685750254966006/16653128209, 25505310154118942544831392038729067903755497221044259/2149036235986823)
(79288775549715152823832461359398/8369449, 39109240493859788707413911769880629277778512201/24212815957)
(-7063162243354980062582366, -46999877893320663260399703093997064867)
(38373428215917266226924496322284534/3995377681, -851010584362506258055921946706932283815295402470083/252543827838329)
(-922743567392548850304724774972244/76615009, 28774490579724626382668911111955091766304373598991/670611173777)
(-4297847514165959495411594419908938/260144641, -5572228242459733050532278409188498850651011129563/4195872914689)
(25693823873572829308390503981657886/6799156849, 10932525244415905436238563886337946791227021141089789/560638076297993)
(1734020861094651845518161190342579621/490576969744, -7048663114128752511391337731329465183281718782416118551/343605996532334528)
(-4919666575019428834163612991825374/1207354009, 1817855779352831733737175913384877974261678062951799/41951929750723)

Commentary

This curve was found prospectively as the specialization t = 3726/881 of an elliptic K3 family with generic Mordell–Weil rank 17. We found 27 independent rational points on the specialized curve, proving rank at least 27 over Q; the exact rank is not currently known. The submitted equation is a globally minimal integral Weierstrass model. In this presentation, the specialization has 10 independent directions beyond the 17 generic sections.

last edited by Roy van Rijn at 2026-09-07 08:27:41 UTC · history