curve #7

y² + xy = x³ − 431092980766333677958362095891166x + 5156283555366643659035652799871176909391533088196

a-invariants: [1, 0, 0, -431092980766333677958362095891166, 5156283555366643659035652799871176909391533088196]
rank (lower bound): ≥ 20
conductor (N): 73813242020125452593092920037241715783942979322145629022774016712292126930
naive height: 237.8517
Faltings height: 17.8502
discriminant (Δ): -6358347962741427332351207823555533236599280133229903044072065747055673142684922784840133195530240000
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 29, 53, 1759, 539449, 1884347, 78324820513, 388882789386500953248084998144029301891
regulator: 2975173777358668583558.17709962338365235667750391729197206924733709891658896356
submitted by: David Renshaw
last updated: 2026-07-01 22:30:39

(1117677105220842826524/37249, 31530479477185489011505872316434/7189057)
(38095017214360176, 6634638907482675334232862)
(128263157005359747/4, 39438837388807975937649915/8)
(173541370721241727764/4489, 2045813113492578321709774985406/300763)
(114037038978699019879860444/2903808769, 1093029826650184196976652135696199191086/156477543135103)
(102579683196689625565576980/3236130769, 889405931520755349254783091883555261398/184093771056103)
(520590665688949735068/11881, 10865365165484759274005818215450/1295029)
(201537570874848579/4, 84414630327852273660698571/8)
(8566017671075667672/169, 23408663211165662031648247674/2197)
(84810811649507676, 24054695979596704444705362)
(-21830796739843140, 2040388505636168283880914)
(-2234086367006310516/121, 3476314228926730107073128678/1331)
(-398890292913112314601476/47513449, 936915725382816974616861962133589434/327510203957)
(-38850378311984740900/5041, 1013647136758546790991381788254/357911)
(410916153652874282067804/58874929, 712483344051989593825064319402912354/451747330217)
(3030869760973710007623516/266375041, 5708794986061809828924957672204713682/4347507044161)
(5668123803956059068/361, 10307638984731401904281889030/6859)
(-5809361085179727432048324/267289801, 9018279752358533631568966242553347738/4369920956549)
(-256381598399113962133604/10169721, 1315536690951996543879087852628778/32431240269)
(479228870284501996956/167281, 135888316201098799476616096547298/68417929)

Found by Nagao (1993). A historical rank ≥ 20 record, via Dujella's elliptic-curve rank-records tables.

last edited by David Renshaw at 2026-05-27 22:31:39 · history