curve #168

y² + xy = x³ − 17499693052051780686008631x + 24444545444646300140928746272873582761

a-invariants: [1, 0, 0, -17499693052051780686008631, 24444545444646300140928746272873582761]
rank (lower bound): ≥ 16
conductor (N): 100907360654491808490244547862425873324272623451710
naive height: 185.9863
Faltings height: 13.4868
discriminant (Δ): 84846485315177942827767106512600101143726929143544805106516597149399675699200
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 31, 157, 821431020607251773, 2908286759476644848543
regulator: 512847700234413.5846609390960665788755182046399204228090518568879266448
submitted by: Seewoo Lee
last updated: 2026-07-01 22:39:41

Witness: 16 independent points

(8749030777802664/2809, -58499245138940147954151/148877)
(9742749700101321/3136, -44196140114557026187593/175616)
(3123962677398, -513183191869000203)
(5367228943899126/1681, -72934916594247767556723/68921)
(5547244131698613186/1692601, -3335257317853511364588473523/2202073901)
(329295175913651664/94249, -70598302430522311861725759/28934443)
(946750469202830567910/247464361, -14298856189125828821136719305689/3892861862891)
(22196650584696645/5776, -1639371507717248414470239/438976)
(312234688643535594/69169, -111299518938285523319516481/18191447)
(1414957717465494/289, -36808103808527122498779/4913)
(9251504323176, -25580962158180935973)
(45938210681459/4, -292653878256911864789/8)
(2429484162389272848/11881, -3786002067932057242550465007/1295029)
(4031610336891147702/5329, 8094894445441753966286486445/389017)
(138398804253815111544/6436369, 1599095523484338264148228166931/16329068153)
(621178546558602/49, 14712769473687372255771/343)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=13506/11: 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 16:07:49 · history