curve #3488

y² + xy + y = x³ − x² − 1433531895173908646196681320514467x + 3194140944286598406642236152903250797239004587875

a-invariants: [1, -1, 1, -1433531895173908646196681320514467, 3194140944286598406642236152903250797239004587875]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 838256123569356158476120539959505376633930110489000273931854
discriminant (Δ): 184131875554992201997692952405763209609933169191617009997844541112566444653561646552768590063748513792
Faltings height: 18.1210
naive height: 240.6500
primes of bad reduction: 2, 3, 11, 23, 29, 43, 67, 71, 89, 139, 199, 443, 499, 1579, 1777, 3547, 5381, 28109, 32933, 37537, 70313, 435769
regulator: 21142833061760751175450.68075425453854663113900149660906566837800598836
submitted by: Steps Unbounded
submitted at: 2026-09-27 10:22:17 UTC
last updated: 2026-09-27 10:22:17 UTC

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

(-489872323568834203588099537947/25260565428121, 618085550511808721916249113080237827648605846/126959323975516436669)
(-196804680242789310024490313613/5342445145129, 30277896782420925037576198996022580519037178/12348383462432252117)
(-192451538759528391534735421149/5029052928025, 15995221527906642721530104430938449020843286/11277927789007103875)
(-131261542629828056605022216829/5955049448209, 71315329460188121199044419994506399701348970/14532089303316078073)
(-2211905557494967498340927541/121958100625, 205267201325802795899531620319336752733602/42590817690765625)
(-53829714873983456912592161/2639390625, 663569004244075824489765421279005876062/135598693359375)
(-1984175308043758987891605/7164437449, 1149182230107944206721316697045439507230/606419478995707)
(-3417277138243208330477/114921, 172159789463453030053321705415846/38958219)
(2670041495568279276603/58081, 82151565019589559362009146943314/13997521)
(8466622243978127913795/13697401, 77019708956088551664154817490477194/50694081101)
(88499228762503556458599/2319529, 7101313109247607422064081836407354/3532642667)
(921729424784234335750123/15437041, 692793113651409585159397995521727426/60652134089)
(1683717592394279479168389/40998409, 967597361720882568360327617062241722/262512812827)
(11379207641570642334462123/129937201, 34713763354578033380817747640445738366/1481154154199)
(22758324081466955004745945/345625281, 89566178643780177313308197999172000760/6425519599071)
(16516673743929045374803029339/16480654129, 2121164853409254018722912241148568460965506/2115736935118633)

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=11/5, t=-529/1848. Found by sweeping t=a/b with the lossless conic-Hilbert prefilter of Breddin (Zenodo 10.5281/zenodo.22242109), screened with PARI ellrank (2-descent): 16 independent points, 2-Selmer upper bound 16.

last edited by Steps Unbounded at 2026-09-27 10:22:17 UTC · history