curve #3206

y² + xy = x³ − 367459594832743192121564798503834635x + 21984661761019378634495434090738640299893358837979121

a-invariants: [1, 0, 0, -367459594832743192121564798503834635, 21984661761019378634495434090738640299893358837979121]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 7210387210131632436261753944550733665686372892278111682534
discriminant (Δ): 2966678824492889969226683679546865785807244790410849948256562594250559795589427442837666952195201464713347072
Faltings height: 19.5049
naive height: 257.2894
primes of bad reduction: 2, 3, 7, 17, 19, 23, 29, 103, 107, 113, 241, 311, 367, 569, 743, 2731, 4027, 18539, 30931, 83891
regulator: 8526995337068307.9000731223937026900529113241711794800465552992841143
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:42:37 UTC
last updated: 2026-09-26 20:42:37 UTC

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

(-500948227073954754, -283461415913825633855972799)
(618831961967068686, -177686449752803041027303839)
(354928123930978469646/121, -6545314995386561940741799315149/1331)
(-2206972301687051310354/3721, -40146036579517558974806417628507/226981)
(96978527597450937582/169, -19794967659002871547033313259/2197)
(24133730276980682677278/22801, -3115533005963119926120453130213497/3442951)
(26884132496771856140286, -4408024915159726933256439397565439)
(9176727235549505573427883532176446/1519174022355889, -874693212297438935973657704142944295625697876370825/59212212373798163147287)
(-62297171251806603124327393276525746/265374217511812201, -42204302612827000938418408024464108397522795494479445/136706161739364087024127051)
(4002066240510226668213767537520/3513379108801, -6848347445308257354081202217984738422231299849/6585481314915703201)
(2350416733051927950115826110434201/2659406316802624, -85381085358082707435273755345112637793036795986215/137144073214723375789568)
(699961735935344351725016081120/12034509108241, -1187390285943810459533203811514224168328759761/41748662822707580039)
(927626820258076427035790768764821679587536769093296430/901877235416925889645906354909317481, -732863137447335689322724697343858529212680408266021158446740446589129931914847541/856487713579780479973721374503834453454776890490994379)
(270863294703076866521299118006000296/68870806034085441, -139310777790545108701048182959835028974454064842628783/18073936358253200779438239)
(-867150520287461633971046341666445590678146/3207346736264459038483249, -1830632199455553382055683577064412216922741341445055656264538729/5744058694869399174638953537009774807)

Commentary

2-isogenous image of D-z2old141516171819-046, rational kernel x=5073086876276604943/4. Source: https://dujella.github.io/tors/z2old141516171819.html. Independent witness points transported by the degree-2 isogeny. Witnesses saturated by exact rational halves of published and transported relations; quadratic-character independence is certified.

last edited by Natanael Wildner Fraga at 2026-09-26 20:42:37 UTC · history