curve #1923

y² = x³ − 1258789696984823822217267987x − 6539883575070523676445882028347415141534

a-invariants: [0, 0, 0, -1258789696984823822217267987, -6539883575070523676445882028347415141534]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/2ℤ
conductor (N): 425237262681943850014675499511718426424003280
discriminant (Δ): 109178821322378504844239369490466518426651065556313973548490158716964663972398720000
Faltings height: 14.6266
naive height: 198.8134
primes of bad reduction: 2, 3, 5, 7, 13, 17, 19, 23, 47, 53, 67, 71, 149, 337, 433, 509, 673, 1277, 2371, 18301, 178643
regulator: 3030355423531.707469599057622555159431703673744976899434751569400
submitted by: Alvaro Lozano-Robledo
submitted at: 2026-09-25 19:15:47 UTC
last updated: 2026-09-25 19:15:47 UTC

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

(6762411620205583933562849794220341/173316674835086601025, 139627201832008942569458888829756866995962732723264/2281710402751994423359596582625)
(-37771279707539025771775904719/1234810393337161, -2509162913551666630699579518264824436207620/43391075461706310371909)
(-21080598563627159605493/825355441, 2243127637469456405872351240269150/23711636464489)
(-343267288223329061230706990645203573847/11615335656166907627319184, 2756946667896615214949133305444728232898064093608756057195/39586549301927231135057504741061714752)
(-5669866047523937810865609923/692126701506601, 1033622920829027772115803277883249812862400/18208677593372812594901)
(-29736036300610, 67808421414293907256)
(89917014817627, 779268133219662327150)
(-116460803682606936803/15217801, 3053291997700970875746901105920/59364641701)
(231307433035257016124353/4270230409, -80975492410017508893713689221654180/279046746536923)
(489361355289333371368237/28718881, 342328899307000461494294874513110400/153904483279)
(49047278262929017, -10862311552622060196886740)
(-9034704834083070827/316969, -14031547388947139104442450880/178453547)
(221976961600867, -3263691144131794179090)

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-7 family with u = 28/5, (p,q) = (3,7) (S. Kihara, Proc. Japan Acad. 77A (2001) 11-12).

last edited by Alvaro Lozano-Robledo at 2026-09-25 19:15:47 UTC · history