curve #533

y² + xy + y = x³ − 1266487529831865779392531193x + 16706282943794669784408821698768347376056

a-invariants: [1, 0, 1, -1266487529831865779392531193, 16706282943794669784408821698768347376056]
rank (lower bound): ≥ 20
torsion subgroup: trivial
conductor (N): 241154232037315742245497393361765425260101143681401714404719114163883110
discriminant (Δ): 9440634675009827659953833972221002806840631527976842381659765887969239103791772500
Faltings height: 14.4942
naive height: 198.8317
primes of bad reduction: 2, 5, 7, 11, 313187314334176288630516095275020032805326160625197031694440408005043
regulator: 155443454521690292548.251978783544150622523990963507996049669303758391401770483
submitted by: David Renshaw
submitted at: 2026-09-02 19:20:13 UTC
last updated: 2026-09-02 19:20:13 UTC

Witness: 20 independent points log in to add more points →

(2255485070205, -117733655455667470018)
(23788512122280, -6339477845192470393)
(23724977687085, 3616766059212129332)
(-737235888715, 132814089618736199182)
(-1273438365140, 135340355623518053132)
(-265605374360, 130547501452529672327)
(-386197568460, 131131002429369017927)
(17229349667985, 259477774646323632)
(-40423220401330, 42997228319011052882)
(11926795629895/9, -3310145129730530373476/27)
(767377433183815/9, 19630108388808896490164/27)
(981625649750815/9, 29280741825455766472864/27)
(7812539722725, -85373555081886647508)
(503620295655, 126761909857285229112)
(-827959599705, 133245322551617971632)
(44179380581297, 216757316807814585598)
(-40913945550155, -5945326195157340868)
(102758596850215/4, -268298927782570034219/8)
(9183698227070695/324, -348989827653634027580351/5832)
(917099127054661/25, 17505620962585368804676/125)

Commentary

Generalized Mestre/Fermigier quartic family at integer Mestre parameters (u,v)=(2,6), canonical primitive root sextuple [0,432,707,1426,1606,2075], specialized at T=2564/5 (equivalently roots [1041/16,-517/8,167/8,-385/16,609/16,-565/16] at T=641/20). Selected by a native M=6000 Nagao rescore of a wide-denominator (n<=2000) sieve of the rational q<=3 Mestre families; the 12 family base points plus a quartic-model rational point search to height 1e5 and an integral-x search to 10^12 gave 20 independent points.

last edited by David Renshaw at 2026-09-02 19:20:13 UTC · history