curve #3699

y² + xy + y = x³ − x² − 23830888577439925303865477591x + 1365659408068526380967008026715318015465071

a-invariants: [1, -1, 1, -23830888577439925303865477591, 1365659408068526380967008026715318015465071]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/4ℤ
conductor (N): 2898874751395323625628640249570894
discriminant (Δ): 60474035373840720764585582513008201747431479490337616287680365173087062571193468630784
Faltings height: 15.2260
naive height: 207.6359
primes of bad reduction: 2, 3, 7, 13, 17, 19, 29, 41, 79, 127, 503, 719, 1823, 3203, 4139, 7507
regulator: 1766168846041.514749112525598030497283895477423272269193855
submitted by: Michael Rubinstein
submitted at: 2026-09-30 01:45:38 UTC
last updated: 2026-09-30 01:45:38 UTC

Witness: 10 independent points log in to add more points →

(112176888463691, 322458722065248047190)
(196780058095177643/1849, 14844143459752461270888138/79507)
(197156333848190579/3025, 49724460398598165849914082/166375)
(28517255250582945155/116281, 127080430442754867021929090502/39651821)
(1176604429085674503587/6806881, 27575849632554022051156653147726/17759152529)
(1507397277360757208203/8346321, 41434081354984450154097138198742/24112521369)
(6676796634583870757463/90573289, 84060052034613595371958631311034/861985991413)
(2242538432800383372378347/2098189636, 3324909101635308272956416735069514779/96109674466616)
(4095002777984111220139739/55031998921, 855415120030806202197295491972761670/12909901594878469)
(155934633840944574971056036155/1482158294418481, 8649217284132035388085580809182897924095078/57061329084582866089129)

Commentary

Specialization at t = 1205/4264 of the Elkies-Klagsbrun Z/4 fibration E1: y^2 + a*xy + a*c*y = x^3 + c*x^2, a = (8t-1)(32t+7), c = 8(t+1)(15t-8)(31t-7) (arXiv:2003.00077, eq. (5)), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-30 01:45:38 UTC · history