curve #2159

y² + xy = x³ − 869442372607181209924740672178325710x + 312039315353737479334542068825037861026717661980383172

a-invariants: [1, 0, 0, -869442372607181209924740672178325710, 312039315353737479334542068825037861026717661980383172]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 436526112602665716189006848580942310302569667141197757407610
discriminant (Δ): 11355063526709325849038916734316403461532095374556044261684482877054357859534218510244730543030728000
Faltings height: 19.0452
naive height: 259.8731
primes of bad reduction: 2, 3, 5, 13, 43, 107, 113, 179, 193, 337, 647, 911, 1153, 1301, 1303, 4339, 5087, 5419, 7873, 30431, 66089, 84751
regulator: 25075641006713157.25113783271085298226369106055236796583518244443233252
submitted by: Steps Unbounded
submitted at: 2026-09-26 14:14:45 UTC
last updated: 2026-09-26 14:14:45 UTC

Witness: 16 independent points log in to add more points →

(538383674910806924, 50165775054255444925898)
(26441041488903256268/49, -553463077012736148528022906/343)
(13581742425546099554/25, -783683328669250154399048018/125)
(545056052418721004, 8547805355344900163345018)
(54993898392253985882/121, 138098400473955868338118610770/1331)
(278905922207866312616/529, -171217324404837678377876525174/12167)
(538460681154121904, 148309170205384797687218)
(283180197713830304, 297554973797809434187897778)
(70451046011360544057074/137641, 1705394832664134337715191675699528/51064811)
(441387485472200528954/841, 416900086690607916630934005232/24389)
(26379125705400375926/49, -1035874323538116921930436/343)
(112590008467042643769704/32761, 36512408765445421798913757564129518/5929741)
(687813124007374988, -198546758289344993562982294)
(-589020799040310586, -787274310063905079961325452)
(110013682072007102718564908/204404209, -473294568288278918151943888517345846/2922366976073)
(223561313921671574, 358941715011215512577681798)

Commentary

Z/2Z curve from the Elkies-Klagsbrun rank-9 family y^2=x^3+2A(t,u)x^2+B(t,u)x (Elkies-Klagsbrun 2020), fibre u=2/5, t=-688/2887 (the 2-isogenous partner of that fibre). Found by a Mestre-Nagao sieve over t=a/b with |a|,b<=30000, screened with PARI ellrank (2-descent): 16 independent points, 2-Selmer upper bound 16.

last edited by Steps Unbounded at 2026-09-26 14:14:45 UTC · history