curve #4593

y² + xy + y = x³ − 67538080824390112575763510074x + 6748853179374408887179706582485861189763116

a-invariants: [1, 0, 1, -67538080824390112575763510074, 6748853179374408887179706582485861189763116]
c-invariants: [3241827879570725403636648483529, -5831009146979494141265085843355889522928057733]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/3ℤ
conductor (N): 91936250668693485040694250195331770
discriminant (Δ): 40019592367262252343616815259828557179018457598565730257057187558353641049217401275200
Faltings height: 15.3374
naive height: 210.7611
primes of bad reduction: 2, 3, 5, 7, 13, 23, 29, 41, 43, 73, 101, 107, 443, 4793, 33941, 503711881
regulator: 199210531841.1040387537954730681615465841522363731488471725
submitted by: Maksym Voznyy
submitted at: 2026-10-18 16:19:14 UTC
last updated: 2026-10-18 16:19:14 UTC

Witness: 10 independent points log in to add more points →

(110185704745216073/256, 31036721749900850323751763/4096)
(18218821702492396910/116281, -4566052620006680941360821048/39651821)
(331316742348758, 4554256605405388339953)
(3127037760156867529142/8392609, -140325320531958329634091905417096/24313388273)
(142521235750284122/289, -47475202610558299011094011/4913)
(14990557472021154992173/56085121, 1172432844673907358969685266902297/420021471169)
(142066981626105612197/908209, -93782241655772155813253048799/865523177)
(2955494854406019557318/7946761, -128831323526349264043803753466008/22401919259)
(76073472201899258, -20982021541818477937336587)
(448796484060241839284191793333/2831017943850721, 24365890261879737187731052761969957122989143/150630852185045849821231)

Commentary

t = 61/382 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 2026-10-18 16:19:14 UTC · history