curve #170

y² + xy + y = x³ + x² − 1002421447003943457225x + 12481234946887000688835790191735

a-invariants: [1, 1, 1, -1002421447003943457225, 12481234946887000688835790191735]
rank (lower bound): ≥ 16
conductor (N): 5859468212246645188748237684777485444087182008624730
naive height: 156.7267
Faltings height: 10.9592
discriminant (Δ): -2831445042543812469319980404169211270210294978426890329088000000
primes of bad reduction: 2, 3, 5, 11, 13, 717496229, 2773745671, 686301029029469989491869608943
regulator: 755715242422743.0255065980163703580980962319872380216750682669847423293
submitted by: Seewoo Lee
last updated: 2026-07-01 22:37:40

Witness: 16 independent points

(751898661471/25, -386048269516272384/125)
(316496094584587/3249, 5364912463885590522304/185193)
(21289729305823/81, 97551108696840416632/729)
(373302054943, 227287226501355848)
(7617130217783727/12769, 663876097852519025472376/1442897)
(111751965933, -36001073991025792)
(54441827983, -10921034534816392)
(49313028731983/961, -292136740323616231672/29791)
(5650946185995727/127449, -338053143254526723960056/45499293)
(9552398586646567/499849, -194798114168374305739256/353393243)
(41878570208387/2209, 56001451768964000304/103823)
(25153429663, 1783620616336328)
(1427445371977/49, 970238560834856594/343)
(272873933446253137/7360369, 102353737160916667605018296/19968681097)
(33618451033, -4095985604896642)
(8651547690714212467/123587689, 23208991631678744562007616404/1373924338613)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-498, -216, -6, 414, 552, -246] with shift t=771/5: 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 16:07:59 · history