curve #5996

y² + xy = x³ − 51403064072894560734652865x + 181174189583993508239587546409315481225

a-invariants: [1, 0, 0, -51403064072894560734652865, 181174189583993508239587546409315481225]
c-invariants: [2467347075498938915263337521, -156534499800574092139616888506021470784681]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/3ℤ
conductor (N): 13248616279429857526445142990382290
discriminant (Δ): -5487459592656539598234556515135205131426620429422979609850314651489088000000000
Faltings height: 13.8191
naive height: 189.7082
primes of bad reduction: 2, 3, 5, 7, 11, 13, 37, 43, 53, 73, 113, 8039, 7172542378144321
regulator: 2365993945540.24049752898635012123959560785193135532960163275
submitted by: Mahesh Panchpor
submitted at: 2026-10-27 17:43:11 UTC
last updated: 2026-10-27 17:43:11 UTC

Witness: 11 independent points log in to add more points →

(87062806349166744010/24039409, -768503751152504026624201931075/117865222327)
(-17568678489248604673880/2463434689, -1663157548796893782648621080910605/122267653919137)
(1487187171265288630/337561, 1243844093748788865637193735/196122941)
(36626410903347177253018/3490091929, -5822661043353338635701436563982337/206184160889533)
(6920649291580, 12525921019892422585)
(232387829516603116/121, -112025542027932350086907503/1331)
(6938830659422170/2209, 739277395813920793720805/103823)
(184042956228442330/22201, 59611240765015457883694415/3307949)
(42533172040655728330/9928801, 196840083497747150011148142085/31285651951)
(372005096860330, -7173703760385277814165)
(3464957170961935498/49729, -6417308831026743875780313803/11089567)

Commentary

t = 1565/17 Family y^2+(7t^2+309)xy-91(t^2-1)(t^2-25)(t^2-121)y=x^3 (Z/3Z K3 family, parameter t=1565/17). Family construction from the Layr Labs Elliptic Rank Challenge public repository, submission/board/3-r12-z3k3.ec (commit c8f82607a219df846ab80dd9e14ee09a2904d709, 2026-10-17). Not itself a repository curve: this t-value is not in that file.

last edited by The Wren at 2026-11-01 22:48:55 UTC · history