curve #2721

y² + xy = x³ − 1404224625110760687046304418687309920124646058907640x + 20253635531680617684878498782482494638392485445816342546348423621425806959392

a-invariants: [1, 0, 0, -1404224625110760687046304418687309920124646058907640, 20253635531680617684878498782482494638392485445816342546348423621425806959392]
rank (lower bound): ≥ 4
torsion subgroup: ℤ/10ℤ
conductor (N): 465432724185682885970875984683390
discriminant (Δ): 171667838884538273419042671054119499692229395395414093164193308574069369346159466907191742659542800695303049038727193007621720073944826985718900000
Faltings height: 27.8219
naive height: 364.9276
primes of bad reduction: 2, 3, 5, 7, 11, 19, 31, 43, 61, 281, 1663, 3019, 101159, 913829429
regulator: 84054986.12375643244574374213703130227394729291
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:19:20 UTC
last updated: 2026-09-26 20:19:20 UTC

Witness: 4 independent points log in to add more points →

(4541052312445649565550702379938733/208167184, -4347141328674606170092020527091849549791989758139/3003436130752)
(9634825434006232487125421644277163430410093523046120666/438160222462278765651980401009, -829972605598108603917541990246483148648135259900944047475346711533671730436962174/290034340881005502277800486723328668218960777)
(12511165309045337662878782934913922130146766494516695591710465341925845384809198175381470210607102432227615860035837491808552349887125170540322955306158294743975283363283061797580719960311/575208145429843294201829059191943864728957574714518379900750294665599611319924446677345925559870917647717683755197854455555156507105905058259831474173512394978404, -406777212681082144988154419380598628312549115676141107962686898236099604343601043299453371818931905219768445229248150671124636762387409768283922337559588441289913148390339646851258581541018100968357228606346456462059338356249828991321539178208116392789478976587054162360175696481/436252110587491091954646404080333091813438407349517463917485847324138342069580387397250674512109247938281415167156289332118014207974401484933887751869171460928496730898270981333481842048607019050916546334074423999800080299660354241045523165208)
(12205418912651372509317738292422313864384515557305314614051245458139038284/564094612319404242590850464394840174706060493521, -7203234375051574914212748139841505075499988337540583334743877298970854159021135394484642326207319256009968/423670209653486612120790401160727523390286083688990668399689456404841319)

Commentary

2-isogenous image of D-z10-025, rational kernel x=-10817324679410223564373776. Source: https://dujella.github.io/tors/z10.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 20:19:20 UTC · history