curve #204

y² + xy = x³ − 40816112666798768214524743126x + 3175238037154846187302797412986141032805156

a-invariants: [1, 0, 0, -40816112666798768214524743126, 3175238037154846187302797412986141032805156]
rank (lower bound): ≥ 12
conductor (N): 8207915399785028827911225140199659880348230953515903817573863786069230
naive height: 209.2511
Faltings height: 15.1799
discriminant (Δ): -3623222374799711246641358701859136275557470213922538013251076487946923600073999257600
primes of bad reduction: 2, 5, 7, 17, 31, 157, 1417178528658108860031670181482848799986917717765832284580576751
regulator: 51826916814533.727636603725739418132382839118508075688787171474
submitted by: Seewoo Lee
last updated: 2026-07-01 22:47:35

Witness: 12 independent points

(9770649366740959079884/619369, -965716610723945297620503113535234/487443403)
(60233506353420124/9, -14776159007078725940508866/27)
(1632143825512484940100/3374569, -60920056763901273997043006329398/6199083253)
(28290587106843356/361, 4641255380757530029904158/6859)
(110391515971620, 121441937340674347746)
(19680454388442956/169, 80229753769020194626786/2197)
(17387195299617109/144, 147427941541656072822491/1728)
(150037170137836, 654838533101906691642)
(244244363437608, 2788656466736583932670)
(5484612139645565/16, 347390158672825672194889/64)
(22903392289503081044364/49042009, 3184412100432593479561128806746734/343441189027)
(419339596095967/4, 1761051162359423345641/8)

Commentary

Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=787/7: 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 12 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 16:10:44 · history