curve #2009

y² + xy + y = x³ − x² − 800367069992061226796029580x + 8688502747437131114604641152964562876047

a-invariants: [1, -1, 1, -800367069992061226796029580, 8688502747437131114604641152964562876047]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 224593554269612009864883936329730274909323150
discriminant (Δ): 201411703273443894919231536176267832442469738144720512622234592951491690496000000
Faltings height: 14.2708
naive height: 197.4549
primes of bad reduction: 2, 3, 5, 11, 17, 19, 29, 37, 41, 47, 127, 139, 197, 3319, 3919, 4441, 499267, 1076903
regulator: 20623821025596243.47188779204192799716381536568774874736016984508
submitted by: Steps Unbounded
submitted at: 2026-09-26 04:59:07 UTC
last updated: 2026-09-26 04:59:07 UTC

Witness: 13 independent points log in to add more points →

(123502737603530077/5929, -14827866757886810632762407/456533)
(349910127263512166710043395301641835085049617585070098252548008084008885468568628954665/13176845318758707920950578242201416207122710384765709473005968974628309361, -3754234784904142540851804231206743881715636269613363824879001362654477804838181105251538180168714465287966039971567885406264989793/47831849260950053081770740598383701089644404607911566843784549329451287968229827204846865535823281755468938359)
(572839509607937537419899635984587203539085333/31151492489083862543630472047041, 2389360860506442711274976369010334737638290789251000054490025623765/173867451440408016725242677341279048183530112161)
(58311353884746069, -14080874813845710739535435)
(48171855644737490781529/2240034241, 3997754322744890826895269515506585/106018580592289)
(1470178511439437805836879238705918783132346124466908970472885/68600044051954653617771282794431388108765039329, -667201395811214664840586090598214807870727759335200087547928957608276040445607793106611963/17967456078160005775687787657273610698653743023849575505197957161112017)
(19990100617045, -26023290093394663499)
(43225393123879013523404042169573083827377901549/856561573329084339525029808089721, 7800065635114194249153648535581200705297265831213143314889091213542065/25069050261587156771255174674070916788339903595981)
(43269546123395528375778541/2099360472889, -93649921222004173412286792231752558255/3041799078296911213)
(39481975081408285/169, 7790054391655062622217881/2197)
(4960531706411645877312549/37291858321, -10816304174613207377722714258386926285/7201468052226631)
(2652857088968940518542224685/144661742506681, 23225692741673992402991001482407080935953/1739925039130548501421)
(39055086244264673989701228193/2268965183926336, 370554547441162776257142527894524784165565/108079107047123818364416)

Commentary

2-isogenous partner of curve#1920 (Lozano-Robledo, rank 13). Found by enumerating that curve's Q-isogeny class: for each member the seed's 13 witness points were carried across the isogeny to that member's own minimal model, and the off-record members with independent images were kept - this curve is one of them.

last edited by Steps Unbounded at 2026-09-26 15:24:06 UTC · history