curve #3553

y² + xy = x³ − 2011143219219703736016434920560646866071203640x + 30156900228721860275494249462949799088593535739436291848589235137600

a-invariants: [1, 0, 0, -2011143219219703736016434920560646866071203640, 30156900228721860275494249462949799088593535739436291848589235137600]
rank (lower bound): ≥ 19
torsion subgroup: ℤ/2ℤ
conductor (N): 2155758992663991692041802618212395866297853180989658043025602632708992617830 ☆ record for ℤ/2ℤ torsion, rank ≥ 19
discriminant (Δ): 127728273940991751526831777509461870520562200054193578351524251213119481632970794325550812349344353824763607619550311975640307246694400000
Faltings height: 25.0341 ☆ record for ℤ/2ℤ torsion, rank ≥ 19
naive height: 324.5587 ☆ record for ℤ/2ℤ torsion, rank ≥ 19
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 43, 59, 61, 67, 307, 461, 659, 769, 773, 1153, 1487, 1609, 8521, 84827, 176597, 261059, 410741, 2959403, 5976301
regulator: 5296103457971436711381.191160296996372341965129436930874522854970906654917159
submitted by: Matej Jurasić
submitted at: 2026-09-28 10:48:46 UTC
last updated: 2026-09-28 10:48:46 UTC

Witness: 19 independent points log in to add more points →

(-37963651047014588678720, 7196704294278910878670205728599160)
(-15512066175475223864720, 7590870073051438655443239488265160)
(-5501897083210849781670, 6407453161811704595908706469033060)
(39039345471925779015280, 3337948452717648121050995021999560)
(169761241712095724812912, 67683612829879392119753446349449672)
(-54280914649362211771279857440/8427409, 160133922521183842240085239442291288361390200/24464768327)
(-37057592621652218812903280/729, 20020031028806131792500736359183631480/19683)
(-3386571036006812477951120/81, 4676822880110955868899455649788360840/729)
(594005051740658826162160/361, 35542682133992686800945376812255066520/6859)
(620921688510517710801688/9, 400283767348492203269076525495046744/27)
(5114572894346221122475120/289, 1611537693646591644614233015041636680/4913)
(24276032166241420312732720/17689, 12314752925605815948307160835789037668520/2352637)
(33528660119676186977729700/961, 46716401925549958703693655220114121640/29791)
(43691876631631499884774882/3481, 539148426535818155457600199653883552574/205379)
(66673483685570273436465520/8649, 3126816567599850036925471670810347903720/804357)
(1956306335951545190242446880/32761, 65762306463577993547287148295207360264760/5929741)
(4779494555210410710517341520/69169, 270503476014622137249377179167311735004920/18191447)
(33315290198269273781580294616/265225, 5729430071838979605911176609448033356941344/136590875)
(1061692109765059301501994831600/13830961, 931697938901122072439975935722409524038576760/51437343959)

Commentary

Rank exactly 19, torsion Z/2Z. Specialization at T = 51349/6454 of the rank-9 fibration of X400 (equations as in curve#802).

last edited by Matej Jurasić at 2026-09-28 12:41:47 UTC · history