curve #209

y² + xy = x³ − 29618038071513025683295304897830x + 61324180930341609392295317188644303710246778852

a-invariants: [1, 0, 0, -29618038071513025683295304897830, 61324180930341609392295317188644303710246778852]
rank (lower bound): ≥ 12
conductor (N): 28341832953461142174879989692232359137024812210317425503392357305906630092930
naive height: 229.0114
Faltings height: 16.9558
discriminant (Δ): 38230734935601059553922446496293184814658591376828071969280861189829807789616058154834984960000
primes of bad reduction: 2, 5, 13, 17, 23, 31, 157, 6329, 694987, 462422851227531199, 20539201174156553881, 210944648998482573373
regulator: 1760119853264209.7771865074582412243590679211918621886021916407
submitted by: Seewoo Lee
last updated: 2026-07-01 22:48:33

Witness: 12 independent points

(6103105912463294758/9, -15076912576194046638463529894/27)
(2721078832888637224/81, -4432992434214499160854395398/729)
(39890447057124135690412/1203409, -7865840853234332855580384931422342/1320139673)
(4306516726393068, -116801695188176626466934)
(586432636053005156/169, -40007927829283736740469774/2197)
(3414095341478124, -480513742457692657782)
(491831239431911341/144, 4806745636338001659536669/1728)
(104843902700829269/16, 24671933214190445149476127/64)
(845855098895920330636/100489, 20357579809465838424373863781554/31855013)
(9980933142991084, 871779419766022779836298)
(13084656463879404, 1383467833157765669272458)
(7989839137371514742387916/387814249, 21864064605954559349916148628421134226/7637226005557)

Commentary

Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=5485/16: 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 16:11:07 · history