curve #4

y² + xy = x³ − 209811944511283096494753999485x + 26653992551590286206010035905960909459942897

a-invariants: [1, 0, 0, -209811944511283096494753999485, 26653992551590286206010035905960909459942897]
c-invariants: [10070973336541588631748191975281, -23029049564574022388452675835133173395678625929]
rank (lower bound): ≥ 15
torsion subgroup: trivial
conductor (N): 164007785374008600847294301252891222936905925387461785510
discriminant (Δ): 284205064269551456641094292955182094002428922847801912953464227178522579040000000000000000
Faltings height: 15.8718
naive height: 214.1616
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 23, 31, 5451269, 1288878077393, 1443189199877, 39758801179649479
regulator: 197339248536833.87234569574893649111280704605566198247249889501297495
submitted by: David Renshaw
submitted at: 2026-05-27 22:11:45 UTC
last updated: 2026-07-01 22:43:36 UTC

(501737473225534, -6905875723539454906517)
(506962472826784, -7112110805804266149017)
(723528240924034, -15925174720190407416017)
(812523671448034, -19814153331319366764017)
(-177569372855966, 7636180350046204211983)
(-419463909011966, 6392016193970552855983)
(-499080158224466, -2656234672725490419017)
(-489670250151966, -3461360602620746284017)
(-161285098583966, -7503202055979337116017)
(576026995032034, 9845125145592749267983)
(569204876688994, 9573248090546012288143)
(437566790688034, 4315766925018741755983)
(383738756328034, 1627462454566927475983)
(380556863867362, 1386435548400000375823)
(373690874760034, 658176881176342211983)

Found by Mestre (1992). A historical rank ≥ 15 record, via Dujella's elliptic-curve rank-records tables.

last edited by David Renshaw at 2026-05-27 22:31:36 UTC · history