curve #803

y² = x³ − x² − 893211001377324088868409658436161336697176597505301441x + 322318489353257476127105984661383880720198001931577617014493148996108419252695905

a-invariants: [0, -1, 0, -893211001377324088868409658436161336697176597505301441, 322318489353257476127105984661383880720198001931577617014493148996108419252695905]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/4ℤ
conductor (N): 14545260576720879419495082714314736176851942086001467334920374959680
discriminant (Δ): 727981240750541222694605443492506965011952771782563642424519939288105666673830161019217697907939220986042382042763173648900896710312090447323656092919412952268800 ☆ record for ℤ/4ℤ torsion, rank ≥ 13
Faltings height: 29.8801 ☆ record for ℤ/4ℤ torsion, rank ≥ 13
naive height: 384.2936 ☆ record for ℤ/4ℤ torsion, rank ≥ 13
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 23, 31, 37, 43, 47, 53, 59, 61, 617, 1117, 1277, 2309, 6173, 8521, 9811, 29753, 123953, 138433, 94669559
regulator: 2959528483927425105.811682154886994510350220814218174837267225900
submitted by: wgxli
submitted at: 2026-09-18 16:13:24 UTC
last updated: 2026-09-18 16:13:24 UTC

Witness: 13 independent points log in to add more points →

(597612317292471058196192631, 1398622144944316533016655735316987556992)
(694353949816071705410997776, 6072970634448242529201207496469911518217)
(-598632869473801622467375497, 25347526376241932883206773405735436724960)
(-933661432500914851209854752, 18503523487265112186367665783433424344255)
(375160895678322358953429710645/484, 104153933807274778329809795564901718854558485/10648)
(683822993666349167197162774469/1156, 38111706823499015231331988739330510894730973/39304)
(3350076553193143356927710856431/841, 5972709618324560578026964149122537091878422112/24389)
(2659496632285762198527249864536/4225, 845757290666752961117122319549083532947519459/274625)
(8803469959946669150726776232351/13225, 7246709153189091176128601501137042920495475976/1520875)
(59922440315103684935670336679895/97969, 67274167788011522588362862411152926306556728680/30664297)
(39814735278556116776643405054777/258064, 1798384890398849925987320469589374484084700455845/131096512)
(-416171738076975399658856525095187/385641, 1299363047434402764524514232926677865112194116560/239483061)
(38292404270227662119661437613226054707331/65310884695225, 217293391098098252518162242915860479398832158981430051315404/527810894327731265875)

Commentary

Found May 2026 without AI assistance. Specialization at T = 14121191/38310928 of the fibration 'E1' published in Elkies & Klagsbrun 2020, "New rank records for elliptic curves having rational torsion."

last edited by wgxli at 2026-09-18 16:13:24 UTC · history