curve #4009

y² + xy = x³ − 6372841881075050664922288149475x + 6192168707769187478734262562807556408152970625

a-invariants: [1, 0, 0, -6372841881075050664922288149475, 6192168707769187478734262562807556408152970625]
c-invariants: [305896410291602431916269831174801, -5350033763512578440471018291669376611048913382201]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 177599378108948787912335962451915637757725844372530
discriminant (Δ): 385070171935294653403724369825095963123593324453895990406239700940966689255248487299680000
Faltings height: 16.3326
naive height: 224.4024
primes of bad reduction: 2, 3, 5, 11, 53, 59, 71, 73, 83, 107, 127, 463, 1069, 1231, 2879, 5547599, 16379591, 16791659
regulator: 2071941150628864249119799.8038575983208870939563849383440274638655127
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:44:55 UTC
last updated: 2026-10-05 04:44:55 UTC

Witness: 15 independent points log in to add more points →

(282780299513268473649316578734439161081235519512684390/57708545901173672410092564446960152249, -133420193233478310756229640344776457944296379665974618336800879083304365993333125/438389553927203774772394394832307868316330644047859700307)
(358595590609508951658163353967620737283986/89123025448410593563491481, -179843771179779774767371938979473714164086925299299995313710777/841365851878526515483692197623424893379)
(35645657570558281988332248735293657366/21149821002469161499225, -47536755406838789070519581459174469511212786707325189181/3075813822090663229394547792755875)
(983407432099957066760258376133910/562601859110817409, -8369445896301330324166219178690816343788399674105/421989596687152258584523327)
(114993157533636145203301391120/48107444018209, -22671606437977421666509125606760451017593775/333670970660428768177)
(43328422644897540464558916106219729/23952603207350502400, -2831355256738170609341525730577632708984623024673143/117227385796389361463775232000)
(1614440378231709318601617400/866103838609, 22651770260424911414589372508307628430775/806036939089950023)
(5318679503719681136170449660603298/3544770581038868769, -18957784798791141323916399267742648854425383470943/6673938480633492065439534897)
(124375489344633103192029112868/85085920985209, -69802065278515356943183484986527770303671/784849807609527813427)
(365428272827768214236321196381076/247555764671157121, 148625600790764074609675289031069325159659610147/123171311546306469146998081)
(3068653679612883623053762919517653150/2089401598475288733769, -65920620643393750999275466290345871525491529943111075/95506489894009508941255367878853)
(3208599949828481668365954216710/2190738822798241, -39791861958087127294008327248252186063759215/102538251354144370626289)
(354206342430506524680011952983900800/195890558773539723921, 66075110584000948055723801366251931350344315626293625/2741702055096930405573297259719)
(73451827978080570844789033988928346/16710823157981291641, 17159961127774776425320918859652118696109268367099667/68311962946526059953511765811)
(112700819835830192549431611484632504595049596790/76774321591156531537591589437729, -429375052999291300902385638378654511952178570643911699664619839866625/672703955429018044473844666818658140861177343183)

Commentary

Unlisted rational-isogeny neighbor of curve#3649 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 15: 15 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:44:55 UTC · history