curve #1920

y² + xy + y = x³ − x² − 443597216327637478631677580x + 16474660966237211646579344641191673788047

a-invariants: [1, -1, 1, -443597216327637478631677580, 16474660966237211646579344641191673788047]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 224593554269612009864883936329730274909323150
discriminant (Δ): -111664459112571378368425429361656266837613120057253819262564250987250741691744000000
Faltings height: 14.6173
naive height: 198.7284
primes of bad reduction: 2, 3, 5, 11, 17, 19, 29, 37, 41, 47, 127, 139, 197, 3319, 3919, 4441, 499267, 1076903
regulator: 2517556277538.603939439427739493163716286055381805267011739483042
submitted by: Alvaro Lozano-Robledo
submitted at: 2026-09-25 19:10:53 UTC
last updated: 2026-09-25 19:10:53 UTC

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

(54444683125045, -392057881853855455499)
(-3886448811156382549446778539843595056555653/393305365112235770069336810449, 34789414802529491923648404420378513666573021188436919098532403697/246657772989928265522829479493323371006879207)
(1081695808660313571143352008205/227037481599157009, -13012798004505072472888708246722400719336974869/108179841582145972652331527)
(-31166595826611, 5118080424921360205)
(-3634375412433976989/1014049, -137071623535355541924027104335/1021147343)
(4021603197052907703505280811421/69338951244347409, -7871318539209645476600781749455276773936563645/18258533932637022078408327)
(-416382544131, 129070891247409650365)
(-19769439317701006185272498411619/986770691481602329, -129004226468348076201902066475152716102776051265/980221813129172282899979267)
(-26940109601198017379/14615329, 7346171158940552617267379126475/55874402767)
(-28609996001747, -75814113592448041179)
(443755582428374949/841, -295388708300165539455149455/24389)
(4988240985733, -119941702069434398379)
(387479410716229/16, 9036258303565469309485/64)

Commentary

Rank exactly 13, torsion Z/2Z: 13 independent points; 2-isogeny Selmer bound 13, confirmed by a second 2-isogeny descent. Specialisation of Kihara's rank-8 family (S. Kihara, Proc. Japan Acad. 77A (2001) 11-12) with u = (s^2 - l)/(2s), (p,q,s) = (3,-2,29/1).

last edited by Alvaro Lozano-Robledo at 2026-09-25 19:10:53 UTC · history