curve #3485

y² + xy = x³ − 53841083898236582744512394752567955238x + 152040732478837697112301241289765721129521047217147781092

a-invariants: [1, 0, 0, -53841083898236582744512394752567955238, 152040732478837697112301241289765721129521047217147781092]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 3633958535429087440559985546561230089154484056735382962822667450
discriminant (Δ): 2706872560514487077203613393909313786743701775469765738912067598155820590361412155618692440070651048663104000000
Faltings height: 20.3928
naive height: 272.2509
primes of bad reduction: 2, 3, 5, 11, 13, 19, 29, 37, 53, 73, 109, 131, 191, 419, 421, 443, 571, 727, 1097, 2953, 3529, 5009, 28751, 110629, 327499
regulator: 24565558987733038019152.7202217331736510367272780842250582717652251691882
submitted by: Steps Unbounded
submitted at: 2026-09-27 09:39:58 UTC
last updated: 2026-09-27 09:39:58 UTC

Witness: 17 independent points log in to add more points →

(-4802607414307397416462368/1164241, 21900930456404710104591694598978918586/1256216039)
(-6412672703849423502384/13225, 20292926270812982671743346722817794/1520875)
(79778036415196702270113/10816, 14058328124430909411630699345809253/1124864)
(9247500360920298363596343/2244004, 1289443545413676144223126458705211893/3361517992)
(17055575604334797626963028/3988009, 88487536331349967758400616905749642/7964053973)
(1077734389391706310395463842/285238321, 7692101853399726461616000057838556666256/4817390003369)
(1614994576485744722333183148/377175241, 553506387622937569242082474352040264882/7325120355461)
(5507240000019512659764152772/16738925641, 25103248429448149799270650307504461845407406/2165665460506939)
(47000638699673378861884541772/10866603049, 321222750377731705699611002100014538538734/1132767301636907)
(75924507648322840132605774252/10576271281, 12660218325367796210788316090549813465305434/1087674314809321)
(186806501067797048081492313852/44669245201, 1225312108918336100862440618166341123722374/9440889642476551)
(477548328864233486977033084104/38803090225, 281203507561078797192609597434904825646550238/7643626727971625)
(8641053249013966174255122398706/1953870000481, 1782972684410882486102975668815027064626964320/2731137071502346129)
(72849655150974142636675968983622/17666360640769, 27738320813515876751709568165330506619250921070/74254134064559892353)
(4497791835397316855343558730552596/627663381410521, 182161395677868565848488729058482704431049376320230/15724983131451260232931)
(5240980104291669418608357119239068/484157664837889, 308331551486467429448202420231897351071842696612278/10653203363346672156287)
(15165488900656768857840906840162732/7326413123391961, 4410280808249198865392758173263011419775820781782946/627100440971366380663709)

Commentary

Z/2Z curve from the Elkies-Klagsbrun rank-9 family y^2=x^3+2A(t,u)x^2+B(t,u)x (Elkies-Klagsbrun 2020), fibre u=11/5, t=720/73. Found by sweeping t=a/b with the lossless conic-Hilbert prefilter of Breddin (Zenodo 10.5281/zenodo.22242109), screened with PARI ellrank (2-descent): 17 independent points, 2-Selmer upper bound 17.

last edited by Steps Unbounded at 2026-09-27 09:39:58 UTC · history