curve #207

y² + xy = x³ − 3341292704781910718198379200256x + 2359087778502928782734938491871399298406363136

a-invariants: [1, 0, 0, -3341292704781910718198379200256, 2359087778502928782734938491871399298406363136]
rank (lower bound): ≥ 12
conductor (N): 877030824470138245488472257225334285674448305325975329534240593693229985430
naive height: 222.4724
Faltings height: 16.3607
discriminant (Δ): -16816563281216520792334587550778720192475526428103707017379511232474985349250763174857420800
primes of bad reduction: 2, 5, 7, 13, 17, 19, 157, 1153, 931690673197751, 1040689837985892942452948677587572182667636492893
regulator: 296423040996438.87421831030116833563744437043438121358288765559
submitted by: Seewoo Lee
last updated: 2026-07-01 22:47:36

Witness: 12 independent points

(10124204966610246124/9, -32213715633639840416999764496/27)
(8018456777604735778/81, -22701924064273317034280911460/729)
(258291942189627096/169, -63006127372568334146060224/2197)
(87243969374310784555424/78552769, -2963040146283390716652256622504576/696213191647)
(157375705483764349/144, -6177158242078167048846049/1728)
(987942511482496, -4727008292132400886208)
(13951748666332005/16, -666805218776552253396811/64)
(397195526444269578582656/177129481, 184755682663955878491031799200844608/2357416262629)
(10083618951832736, 996976824896800215156352)
(11918542857126748, 1286695214692376067191380)
(6843911934657346548837664/416445649, 17797961083287660114813363226306796416/8498406359143)
(2769496506371736, -119782031103604577330648)

Commentary

Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=8863/13: 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 16:10:58 · history