curve #172

y² + xy + y = x³ − x² − 145932228185273125026647x + 22189248867258609149920496015950319

a-invariants: [1, -1, 1, -145932228185273125026647, 22189248867258609149920496015950319]
rank (lower bound): ≥ 16
conductor (N): 25711696321134843123753387143512024261228018943567970
naive height: 171.6930
Faltings height: 12.2267
discriminant (Δ): -13801249682874560225185953107990803809947314431823775922758046720000000
primes of bad reduction: 2, 3, 5, 17, 19, 37, 904594283708550341, 26425921171690203209101347263
regulator: 492472590511832.3165935628802829899244997577628812687516700439862003778
submitted by: Seewoo Lee
last updated: 2026-07-01 22:38:41

Witness: 16 independent points

(77386157874918547/323761, 5719519573656468731547374/184220009)
(-7462742259149/100, 180732369761751418069/1000)
(72287956542747867/546121, 29080163520921100036204234/403583419)
(20575728680277/121, 63706932078410060606/1331)
(4080864960914165907/22363441, 4277533349985510929005804654/105756712489)
(6220070339342547/28561, 131038128988252422570574/4826809)
(393063052397707/1681, 2010382117608086193486/68921)
(866141278428643/2209, 16497886662035545362978/103823)
(293058398233203/529, 4060260064355711051762/12167)
(2965346037177, 5066018568264726586)
(26499730120707, -136400935028918075714)
(-1934745224044713/11881, 264246600850528066689094/1295029)
(-1427985823453196933/11485321, 7628586664397936536701775734/38923752869)
(-840290327883885983/19280881, 14284229246454023303849298956/84662348471)
(-27104667413, 161631633543294526)
(6309240191833083/29929, 145887234306918928925462/5177717)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=501/10: let p6(x)=∏(x−a_i) and q(x)=p6(x−t)·p6(x+t); completing q to g(x)²−r(x) gives the genus-1 quartic model y²=r(x), whose x=a_i±t base points plus a few small-x extra points supply 16 independent rational points. The shift t was selected by a Nagao–Mestre prime-sum sieve; the witness points were found by an integer quartic-point search and certified independent via the Néron–Tate height-pairing matrix.

last edited by Seewoo Lee at 2026-06-25 16:08:09 · history