curve #4174

y² + xy + y = x³ − x² − 34485332697179609537x + 224402383073056115314473626449

a-invariants: [1, -1, 1, -34485332697179609537, 224402383073056115314473626449]
c-invariants: [1655295969464621257761, -193883658967671651769114417592241]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 760652629382499837073532406180848996486496990
discriminant (Δ): -19129256019360117175069691888990082620036082051120489676800000
Faltings height: 10.4440
naive height: 148.6896
primes of bad reduction: 2, 3, 5, 7, 11, 41, 8753, 305852722779866160643512871771605391
regulator: 2511252938081386296.583470426742340388215830671178595415306361282686808
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-11 22:58:17 UTC
last updated: 2026-10-11 22:58:17 UTC

Witness: 16 independent points log in to add more points →

(-7632680393, 207258880854596)
(-2019046403, 534602851710746)
(57904976857, 13870159463206046)
(420383656957, 272537758012325096)
(-30229700730897833/4363921, 3298056579321221900147444/9116230969)
(-29519496827733473/9803161, 16838029039901403613868236/30693697091)
(-2973375710795897/579121, 227347981775228586602788/440711081)
(-1393874944987793/1692601, 1105965951807859412769796/2202073901)
(-280054675219937/38809, 2387033464165085984308/7645373)
(-10070052520957/2209, 55608516350617458748/103823)
(-926622619073/121, 264142463131725196/1331)
(-713555528641/121, 628097488062045068/1331)
(244555844443/9, 118771503453203692/27)
(4667795361119/25, 10079982803555842868/125)
(24789889062433/169, 123332763971948831912/2197)
(609677274950076157/104060401, 501806972816879294498397146/1061520150601)

Commentary

Specialization at T = 1/9 of the K3 elliptic fibration y^2 + (Lx+B)y = x(x+D)(x+E) given in the commentary of database curve #3593; points from PARI ellrank.

last edited by Andrii Prokhorenkov at 2026-10-11 22:58:17 UTC · history