curve #3487

y² + xy + y = x³ − x² − 1978662656511796880788679245183232x + 19547913950717552283422700164968307960838375127331

a-invariants: [1, -1, 1, -1978662656511796880788679245183232, 19547913950717552283422700164968307960838375127331]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 11691134462264714370561833310976107653897746417708594365510
discriminant (Δ): 330710880843178164195918151868871968794900314011642036913433405619191247033822171587664657574174720000
Faltings height: 18.1788
naive height: 241.6168
primes of bad reduction: 2, 3, 5, 7, 37, 53, 83, 107, 181, 239, 439, 617, 751, 827, 857, 1051, 1223, 1913, 3169, 11807, 36931, 948749
regulator: 47880763636885620546779.34055916719517740013007707216025079195317261860
submitted by: Steps Unbounded
submitted at: 2026-09-27 09:46:17 UTC
last updated: 2026-09-27 09:46:17 UTC

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

(-1254529604465193264042220539/45428233321, 70576009063336172430424579940841457531051/9682528221804619)
(-519973316751840199190433699/65715835201, 99248434685329182798985947006572974448959/16846320069611551)
(-62380407838705440424065275/1278849121, 2549310623344297569964758384605057217/45732923416081)
(-94526129916568439216075/4214809, 62782490584032528842995704346121949/8653002877)
(958999788373029141/25, 38065501105146985685231549/125)
(7660915060355764502425/40401, 652811730754018530823381975880677/8120601)
(12116835140864693952549/7038409, 75033201528302536520705644162331173/18672899077)
(187461572711157771254589/18879025, 76931883450800942761851460706910697/82029363625)
(551200904131273182932389/13564489, 124828381776480885708882444286841883/49958012987)
(1307860295629593573989491749/313442099881, 591575377153125275583731661721447840039019/175483380597276779)
(7699849759858649604862762749/15091876801, 673129543579221746812619703384454610226881/1854021973126049)
(12948728542313534425872794001/1585405838641, 3958433587950859890871054433834709676777121/1996230468202203689)
(66961445347559919321055765029/1233727797289, 11631194609081521313961021113113979192215093/1370342177466202837)
(892607524959670089257944012089/21643029970849, 286445046697058797317050803660503027984481427/100687855531593513743)
(337936358954954371100312215622899/7303882834041049, 3246283855196604566384554867934588539011000774287/624209963802121498712093)
(81450890532247565317120483856452149/1017041105741424481, 19855133054403136806621959157837372920306168481900119/1025670251008512835753088879)

Commentary

Z/2Z curve from the Elkies-Klagsbrun rank-9 family y^2=x^3+2A(t,u)x^2+B(t,u)x (Elkies-Klagsbrun 2020), fibre u=11/5, t=-1621/3315. Found by sweeping t=a/b with the lossless conic-Hilbert prefilter of Breddin (Zenodo 10.5281/zenodo.22242109), screened with PARI ellrank (2-descent): 16 independent points, 2-Selmer upper bound 16.

last edited by Steps Unbounded at 2026-09-27 09:46:17 UTC · history