curve #201

y² + xy + y = x³ − x² − 2033019015003100703x + 1111839781374124406036388487

a-invariants: [1, -1, 1, -2033019015003100703, 1111839781374124406036388487]
rank (lower bound): ≥ 13
conductor (N): 38936103625318215160524810514628790
naive height: 138.0818
Faltings height: 9.3280
discriminant (Δ): 3746479851109552777476494863795034246987040897069772800
primes of bad reduction: 2, 3, 5, 7, 13, 19, 37, 41, 109, 463, 1295809, 48684511, 51807199
regulator: 88401680947.32886924606362766775259075928101595166417470536057740
submitted by: Seewoo Lee
last updated: 2026-07-01 22:45:38

Witness: 13 independent points

(59588989675, -14542085792666722)
(38882274373, -7661970975095272)
(19682641375, -2754329599786054)
(9417396553, -903979327498222)
(3767050411, -216589785741346)
(30967318693/16, -4258458933181603/64)
(47002945240745371/35772361, -5697237047256759529124786/213954491141)
(1094757112585/961, -493347650698187044/29791)
(121707573635009/112225, -507973989441742763468/37595375)
(3615378856393/3481, -2256266340234165578/205379)
(23288579411/25, -641863748313274/125)
(108257172103/121, -4013446770576302/1331)
(749728245355/841, -68704246405359218/24389)

Commentary

Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=939/1: 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 13 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 16:10:29 · history