curve #3203

y² + xy + y = x³ − 4558674843837445810853838462723128x + 69408437613126590888662251760059596464201831554998

a-invariants: [1, 0, 1, -4558674843837445810853838462723128, 69408437613126590888662251760059596464201831554998]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 1753089367555057839284708707858690346751194714173298539554510
discriminant (Δ): 3981941762460561343555472172816100414236359376523536789477216738158867620176959707987757265554031450000
Faltings height: 18.3865
naive height: 244.1206
primes of bad reduction: 2, 3, 5, 11, 19, 23, 31, 41, 43, 137, 197, 313, 353, 457, 509, 757, 1447, 4583, 4969, 104579, 738547
regulator: 91118684888976897.978770343027544276679005892846577046120431950283987
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:42:30 UTC
last updated: 2026-09-26 20:42:30 UTC

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

(15262030261855914, 1840864545737137709689495)
(352750779051217626/49, 2085362420895642710960840050/343)
(42041248770915712041/2809, -316971122126852502649607838325/148877)
(210229268311262224809/121, -3045891797783604097315518549835/1331)
(135511501333868309841/2209, -471358648455483223716890909575/103823)
(4072331329079665291626/9409, -256806120318318675537149161269065/912673)
(33329555545330414946/2209, 209239119844013247857468774130/103823)
(2235633811433047784073/1369, 105616789250808940048772104815835/50653)
(108543656047255533039740454509977053/7182085526882286241, -38324443573032385943717194773663643271260673556529373/19247567831904579415263954289)
(38308491519865608234/152881, 493893133583701027965468803481265/59776471)
(2628353951719829616696976250957491959771239151/469755696431930883374586776164, -2137542266931730473236394884660718682864379983191026455000795636077155/321964569736529517375093048686384153565108712)
(299481643084124912479032269804549/4327391116502281, -2630393743104078750815380186583621282173509404565/284668376024277949315429)
(-605425138218512765863063836/10545441481, -12899875112976728811538015746932714233205/1082921931125371)
(763844944653343835660376/47844889, -276544508928358098236413784704397935/330943097213)
(75143158113151271433432/14161, -20596769432239052962942467794972530/1685159)

Commentary

2-isogenous image of D-z2old141516171819-036, rational kernel x=593026323000703735/4. 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 20:42:30 UTC · history