curve #188

y² + xy = x³ − 50180143404585230743736520x + 138805605621811599445299434551151234112

a-invariants: [1, 0, 0, -50180143404585230743736520, 138805605621811599445299434551151234112]
rank (lower bound): ≥ 16
conductor (N): 3668941577753100929827306465184507435413975438465768915789448790
naive height: 189.1754
Faltings height: 13.6454
discriminant (Δ): -236561593535490259936412512778599347267021742231315444284836282901094195200000
primes of bad reduction: 2, 3, 5, 17, 19, 307, 25219, 1733539, 159637111, 5024333263, 22802691139247, 1542477233890883
regulator: 136960321369372805.8656134456998812155236881189062078900842549168588408
submitted by: Seewoo Lee
last updated: 2026-07-01 22:40:38

Witness: 16 independent points

(825716257841424, 23726316375265516782648)
(5525622329342214, -410744203431915461922282)
(91443162077961936/49, -27651820816198881418299576/343)
(108061543553424, -1120974193470845681352)
(56294831605250064/2401, -12807841415343343821981768/117649)
(152843981101086/49, 1218992269125288065724/343)
(216195502811586/49, 622611228485546670924/343)
(78123844306943544/17161, 4879495862885822492127528/2248091)
(223153900113936/49, 746148161760630044424/343)
(1026041124226967808/165649, 546182001320023067322019704/67419143)
(783388313858485297374/111746041, 13548940718902018465047652299948/1181267399411)
(39614166327566417136/4870849, 176194365587628010721239681224/10749963743)
(20346720026928, 86839628682789329976)
(3460337674856336/2809, 1322068898159854927638616/148877)
(139553690836715272193916304/2627579402361, 1634628281522536737464060336430216707288/4259256287218536141)
(70352264711064, 587207690274157016568)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=1889/6: 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 16:09:26 · history