curve #5360

y² + xy = x³ − 413665801584822309765404042296396x + 3238655647459980222622621149824750963665117393040

a-invariants: [1, 0, 0, -413665801584822309765404042296396, 3238655647459980222622621149824750963665117393040]
c-invariants: [19855958476071470868739394030227009, -2798198479405422942129882387555791135715752472927073]
rank (lower bound): ≥ 12
torsion subgroup: ℤ/3ℤ
conductor (N): 165048118968436491259413491400640254451352730
discriminant (Δ): -881174648137977236354532620412342886556308875437861803738437619494916056573590000117515535155200
Faltings height: 17.4382
naive height: 236.9216
primes of bad reduction: 2, 3, 5, 7, 13, 31, 37, 83, 89, 103, 211, 331, 449, 5016621163, 62909039502787
regulator: 1062940042411803.3845625651514441766912300239641968866638735342
submitted by: Maksym Voznyy
submitted at: 2026-10-21 19:53:29 UTC
last updated: 2026-10-21 19:53:29 UTC

Witness: 12 independent points log in to add more points →

(113681266649304678096121784/9060945721, -132467896919115928182478816466118048380/862502362236269)
(17125948887340984805058252515/1443232211716, 50745737287348151975352075109126943035615/1733821244616169736)
(1919320446165241494104/543169, -540212865267151605226404667815500/400315553)
(4470699294371108675069/35344, -295278347936142586004702733415337/6644672)
(1371097601704745945129263040/102207450601, 10512372539313758216438693528374656422780/32675619749689099)
(37473780091263250801591640/3108174001, -10706627867267012761699134815646192020/173283808729751)
(2433205079272863112574534/211848025, -157961059868409683074714713716860208/3083448003875)
(1724733857917502339534998777811541194/146752713349809071761, -31726707800909992726810703160663818442181275074936294/1777784885935149069868443361991)
(491649357026066136460990447640/76219377797641, 609404765092079462573114909020699098747957820/665423445562568854811)
(21106007004081832172/1849, 5068818094719688616450696432/79507)
(156585082345078744864418/3279721, -57068399952613580140089805378060544/5939574731)
(246927709038243966754088/23049601, -21163628366992124854014287200889884/110661134401)

Commentary

t = 8357/1771 for y^2 + (7t^2+309) x y - 91 (t^2-1)(t^2-25)(t^2-121) y = x^3 from https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/main/submission/board/3-r12-z3k3.ec

last edited by Maksym Voznyy at 2026-10-21 19:53:29 UTC · history