curve #1929

y² + xy = x³ − 31554587619036252163726474540x + 30953329716167731427826379130819734911419600

a-invariants: [1, 0, 0, -31554587619036252163726474540, 30953329716167731427826379130819734911419600]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 96220149085890919907173196968787799179447574703830
discriminant (Δ): -411892130434918539681126771736175915091086745573083244150886380383131028680565112172680000
Faltings height: 15.8749
naive height: 213.8053
primes of bad reduction: 2, 3, 5, 7, 17, 53, 83, 113, 127, 661, 857, 1459, 1697, 1811, 3049, 103903, 13470593, 39386881
regulator: 59517402113265.31811432613353910425135530546795766831119894707079
submitted by: Alvaro Lozano-Robledo
submitted at: 2026-09-25 19:16:58 UTC
last updated: 2026-09-25 19:16:58 UTC

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

(162264700075145369789247938903480/54406500635544121, 2064517176696842748728154582978000198946815593060/12690425684713444302477331)
(-241280978994370, 4951799266737307589360)
(642590686603460, -16613759229510019902220)
(240442402875566300/361, -119866851450237665635707940/6859)
(694510495734335/4, 44332456617034088837425/8)
(3368492710896614359720340/20321357809, 15939998354559322342912839636776731960/2896870519746377)
(-2397919116914195/16, 363822893468879632488715/64)
(2593796877483230, -131907573898463468970190)
(466959242838585741145589590/9887370847561, -169057817463111945080796733386773325296560/31090036753116912059)
(-2711157165151688369653/7873636, -22010587436001327888469000236823/22093422616)
(424609280931023750, -276684089600738209314042640)
(-2506559383701371596471213890760/11470602566608201, -6432295913963036087464432030933945677907049660/1228511846955445707872699)
(-316241269983304, 3050469730823500168172)

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) = (3,2,135/4).

last edited by Alvaro Lozano-Robledo at 2026-09-25 19:16:58 UTC · history