curve #177

y² + xy = x³ − 60022613792933162126770x + 5643374875619607135973539223008900

a-invariants: [1, 0, 0, -60022613792933162126770, 5643374875619607135973539223008900]
rank (lower bound): ≥ 16
conductor (N): 18199850239630741806694632255103221302824470127647536470
naive height: 168.9606
Faltings height: 11.8946
discriminant (Δ): 81438791243922645157611866031099042317832628278473480004951449600000
primes of bad reduction: 2, 5, 7, 13, 14143, 61095704479, 2023834707752971, 11436658788285881147191
regulator: 596245719370995.9770421874122669814731596437173131542981089055244318210
submitted by: Seewoo Lee
last updated: 2026-07-01 22:38:40

Witness: 16 independent points

(5249051824831780/9, 380295550092660091636810/27)
(2031681680938015/4, 91576372616636717841705/8)
(2844244702180/9, 3646219393118000650/27)
(290345255752, 112659987508580286)
(191998494861682180/1520289, -16484070580180227559886050/1874516337)
(128168077282, -7471078717004238)
(29107455924484/225, 22258797552123760418/3375)
(478620130398595780/4297329, 164229744232688785540957550/8908363017)
(1771839391525/16, 1205024210105889495/64)
(9629446541859620/19321, -847256984370131499683750/2685619)
(1107272954531184/2209, -33077154155676376659774/103823)
(1167606137791940/1681, -37690495954067937504370/68921)
(1128217426980, -1172179765340581410)
(115330085540, 15967736869394430)
(33583372397920, -194614399746674405950)
(34375110789572, -201537188908789628034)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=7/8: 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 16:08:33 · history