curve #173

y² + xy + y = x³ + x² − 2834094933042959558723113x + 1836350162256247293173176506110827031

a-invariants: [1, 1, 1, -2834094933042959558723113, 1836350162256247293173176506110827031]
c-invariants: [136036556786062058818709425, -1586606540190417935477519966720894875625]
rank (lower bound): ≥ 16
torsion subgroup: trivial
conductor (N): 29160913962360673855189378606441749566045554275575550
discriminant (Δ): 95309530954517651197679942445741588992280745806742990246093888224000000
Faltings height: 12.7056
naive height: 180.5249
primes of bad reduction: 2, 3, 5, 7, 11, 23, 37, 89, 907, 817719259139, 1216105912003, 36958698339265441
regulator: 344112794184068.5132837095483207356329042306934301174297818860928244136
submitted by: Seewoo Lee
submitted at: 2026-06-25 16:01:27 UTC
last updated: 2026-07-01 22:39:40 UTC

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

(23133397748770775/6889, -3141471217094483134890456/571787)
(3041032595591, -4619620262735245200)
(94677769838153919/57121, -17760474203161028032626232/13651919)
(1407320880095, -796958327588913048)
(2086667697367268975/1560001, -1290511508082994089417956712/1948441249)
(1235676276895, -470187049057551048)
(317369273758475/289, -1080809551567787511744/4913)
(896313884656722515/917764, 1685321961566180587914459/879217912)
(949674970078745/961, 796634291908532022882/29791)
(357533678862075/361, 210056649647529768208/6859)
(1024047828325, 89406907895428362)
(80539355466799265/76729, 2852679029117887766832456/21253933)
(67970791480505/64, 79871140889193934449/512)
(1405423329526175/289, 49876486941519954749256/4913)
(547498095067505304755/11377129, -12802996373789791304733627991116/38375056117)
(209690757267880795/212521, 2350255655177088892822812/97972181)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-110, 6, 60, 44, 94, -94] with shift t=3199/2: 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 16:08:14 UTC · history