curve #203

y² + xy + y = x³ − 6897389822833125395x + 7113465576836136307696346462

a-invariants: [1, 0, 1, -6897389822833125395, 7113465576836136307696346462]
rank (lower bound): ≥ 13
conductor (N): 2840704603917455365782151764722058186
naive height: 141.7867
Faltings height: 9.7110
discriminant (Δ): -859076487022818729321026088704523026210935982019962548656
primes of bad reduction: 2, 3, 7, 13, 17, 19, 23, 37, 131, 177023219, 816206070761378053
regulator: 2084640105.287431943944479547655984090376395520069190176449159295
submitted by: Seewoo Lee
last updated: 2026-07-01 22:45:38

Witness: 13 independent points

(1821696636986571/1681, -77752259635965980361424/68921)
(2651704780659/100, -4297661292794605493/1000)
(11101674264, -1139650601752094)
(6952530720, -543352700813627)
(10172211834471/2209, -28051427564651955172/103823)
(2670689975, -87987691259172)
(15395621611/9, -483250003213552/27)
(26665787397/16, -1006638459099257/64)
(1553029431, -12140319849908)
(248019741134157063/172318129, -29292387555156537782663404/2262020079383)
(208894998531/9409, -76143106562997552752/912673)
(7618219107, 629843965912912)
(71190258388521651/5313025, 18654730438432478327441884/12246522625)

Commentary

Rank ≥13. Mestre–Fermigier construction from the integer 6-tuple a=[-138, 138, 162, -60, -90, -12] with shift t=296/9: 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 16:10:39 · history