curve #174

y² + xy = x³ − x² − 2235680741254477841244774x + 1283259936757609808257953240813254080

a-invariants: [1, -1, 0, -2235680741254477841244774, 1283259936757609808257953240813254080]
rank (lower bound): ≥ 16
conductor (N): 145828484209869499295970073108577927976542470773736770
naive height: 179.8134
Faltings height: 12.7946
discriminant (Δ): 3771450175401594216284632730407117555621648825258951862297267210498122500
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 31, 89, 109, 283291781381707777230208241994647890971677
regulator: 328235901585154.0624787348667814960685903389391649690219209571650604171
submitted by: Seewoo Lee
last updated: 2026-07-01 22:39:39

Witness: 16 independent points

(1286357640157919, 46136250466930114059212)
(341905934694529/9, 6317254430329285246819/27)
(12049952967584, 41521230102586140032)
(5541103781669, 12610637009143550369)
(1247207651143055/289, 41691156313706448253285/4913)
(2888647988814, 4350731104079225264)
(1530043997586, 1201854479511790052)
(30722339887386/25, 78231956856463548194/125)
(438561162327677816/395641, 101720666381642456845702348/248858189)
(944756146061, 119758896370184162)
(1432445502137738039/1592644, -11759942951095767341832359/2009916728)
(370009504648442561/410881, -4168265120033709582603338/263374721)
(909974167949, -48529226780115823)
(4263622168043034/4489, -38798376235741601969564/300763)
(988130760281, -197302981196592103)
(1006411077788940806/755161, 537469664983254488916048808/656234909)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[-110, 6, 60, 44, 94, -94] with shift t=2524/5: 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 16:08:19 · history