curve #4579

y² + xy = x³ − x² − 14830160637914727541943945976729x + 21134460491646867693613527739890379784909256285

a-invariants: [1, -1, 0, -14830160637914727541943945976729, 21134460491646867693613527739890379784909256285]
c-invariants: [711847710619906922013309406883001, -18260173864782890483967390177684139074269266456749]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 4834182615430193767926697907961349712388368645771199790890 ☆ record for ℤ/2ℤ torsion, rank ≥ 17
discriminant (Δ): 15786239236164557107876786021501242993888680609800034575422888562684601869255679900965556250000
Faltings height: 16.8383 ☆ record for ℤ/2ℤ torsion, rank ≥ 17
naive height: 226.9363 ☆ record for ℤ/2ℤ torsion, rank ≥ 17
primes of bad reduction: 2, 3, 5, 11, 53, 83, 101, 137, 251, 463, 467, 619, 647, 727, 907, 1277, 4447, 10039, 10987, 14011, 16553
regulator: 123317795439149552188965.517288856627152635228516019510877111643676121147
submitted by: jesper-petersen
submitted at: 2026-10-18 08:10:15 UTC
last updated: 2026-10-18 08:10:15 UTC

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

(-4408335659738354, 29012647655334870732927)
(-1615342248686829, 202176368362299562476252)
(109739561364771, 139672211700784115321052)
(1271733170241715, 65812110933599231168220)
(53200859178747047400378334/41810843529, 562207415056554768474247607661381537791/8549355852279333)
(-2559687006100854, -205727894089296813429573)
(-2806674368438354, -201614820954744953854573)
(4641064178068590, -228632517395949888704685)
(4165247111125515, -177840463457240921892900)
(608481769513232993661289/228856384, -77386415027150076866713181399213321/3462139377152)
(39850322552088690135624819/15467150689, 10179855484213100164548988194022728351/1923603129738863)
(2742493760764771, 33013315341481888121052)
(2452425182763076733566/436921, 97824445755650154989570846059287/288804781)
(-41363978098205163846/11449, 203354331217799918335573928961/1225043)
(651291357891861318849/6400, -16609484297393250412958909106153/512000)
(24539758433290182211/6241, 75764843096311426459279593828/493039)
(-285890208755559214101530709/114913542121, -8043056166506930454475466877409773490587/38954426730055669)

Commentary

Curve and independent-point witnesses from the Layr Labs Elliptic Rank Challenge public repository, submission/board/2-r17-z2k3.ec. Specialization u = 4/5, t = -641/910 of the Elkies–Klagsbrun Z/2Z family (arXiv:2003.00077). The original submission describes Mestre–Nagao sieving followed by PARI/GP rank calculations. Submitted to ICARM to document these improvements and make the curve available to other researchers.

last edited by jesper-petersen at 2026-10-18 08:10:15 UTC · history