curve #512

y² + xy = x³ − 21257297940878768617442961914104483141758957092350x + 37478276525551427290571152164099380097506849748144449472160083430305579332

a-invariants: [1, 0, 0, -21257297940878768617442961914104483141758957092350, 37478276525551427290571152164099380097506849748144449472160083430305579332]
rank (lower bound): ≥ 17
torsion subgroup: ℤ/2ℤ
conductor (N): 53408729173037795322621188411209581292325482397255882514654677241857822588089265070
discriminant (Δ): 7961585506922791313004835825027915933572411635320264996015765046933489091680934177397242361796970688822027713568669995291592829778587317574400000000
Faltings height: 27.2097
naive height: 352.3560
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 29, 71, 107, 127, 151, 277, 353, 541, 677, 947, 2671, 2969, 4931, 14557, 38767, 112589, 267167, 1613609, 2149909, 10787047
regulator: 109305095885381469157.732721802577654106778872879369878438095695310221309
submitted by: Matthias Breddin
submitted at: 2026-09-02 13:42:09 UTC
last updated: 2026-09-02 13:42:09 UTC

Witness: 17 independent points log in to add more points →

(144322296891428228514254, -5866291890058987393338787410137463652)
(243297607518051195849944584/81, 621834818036347610891936400713350168162/729)
(2418243784716276707426198, -463252619141756858588134051209935434)
(2971888356275556171071804, 742991301155567671971495402215576498)
(3209895184957695125868302, 1522333488980749260389806165434083420)
(274693856958183488086455692/121, -1276754549857430641112298224251242080330/1331)
(-2443386461187350389609846, 8650474994143171672214724517122467798)
(-714519374393459885836546, 7232028657621575159832711855127586348)
(20957742447150970905312194, 93793277917057420840673173682326534088)
(-25706815044539377760499284/9, 233659185026329304285137763537051746166/27)
(1563493829588025065892078332/2209, -495609729124139030974577158109421395729042/103823)
(8482616729370666572110604, -21622341264788175247998099341966287102)
(226779952527906388119748665972764/8826841, -3363521406212902365968820368321702188554594714922/26224544611)
(2976726130020675050225624, -759993586979651869957243448587267342)
(79852456772496183148130204, 712399061055359560129736514065286738098)
(5784467548400663979629668143386296/1949840649, -62370751970463885273312018846642915550986125759186/86099113537893)
(694241932243774094330972076884756/30880249, -17934210402735194686146925905843828576900137884366/171601543693)

Commentary

Exact rank 17, unconditional: rational 2-torsion point, 17 independent points (Neron-Tate height matrix of rank 17; the archived certificate additionally saturates the lattice at all primes up to 97 — the points listed here are the ellrank output, not the saturated basis) with matching 2-descent upper bound 17 (PARI/GP 2.15.4 ellrank, effort 2). Fiber t=-27365/3567 on branch u=29/13 of the Elkies-Klagsbrun Z/2Z rank-9 family (arXiv:2003.00077), found by a Mestre-Nagao residue-table sieve followed by an exact 2-descent-type upper bound and ellrank (Matthias Breddin, 2026-09-02; method: doi:10.5281/zenodo.22242107).

last edited by Matthias Breddin at 2026-09-02 13:42:09 UTC · history