curve #176

y² + xy + y = x³ − x² − 39457403617444107354653x + 2995236809415759824391532391142437

a-invariants: [1, -1, 1, -39457403617444107354653, 2995236809415759824391532391142437]
rank (lower bound): ≥ 16
conductor (N): 6430563811515979376580169926167902717057516739737835010
naive height: 167.7021
Faltings height: 11.8259
discriminant (Δ): 55901631537054437994489290286441257543068900330119374782386280857600
primes of bad reduction: 2, 3, 5, 13, 17, 19, 1003771, 997188134941985254983271503747820602358073
regulator: 1807850498928697.709773358085301962190248467956461960084294927955712660
submitted by: Seewoo Lee
last updated: 2026-07-01 22:38:39

Witness: 16 independent points

(144809072073313, 1742582023374680171298)
(119013030166267/9, 1298204132495509294076/27)
(14453193394137/64, 38320836532964296481/512)
(82737779887, -17233834740637208)
(1625900049957/16, -378511156915988195/64)
(99883404201318583/942841, -1858606599514605068981832/915498611)
(94373439663, 10584301043120488)
(1456550150901/16, 803654378416531621/64)
(118824023111096415/1456849, 31435043139868783893538264/1758416743)
(664284748380235207/2036329, -457969898199541897508533896/2905841483)
(11055538290343/9, -36305826547177738760/27)
(1440208560111, -1712732440197729176)
(9113382082811474679/4644025, -27376005199713547093247578952/10007873875)
(307862534703, -141516189959116952)
(50770409072312721919/1103767729, 1310741387619272185155970466008/36670475260567)
(13914484586320425535951/28771283641, -1520159124306311998316785261631576/4880213902470061)

Commentary

Rank ≥16. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=2913/4: 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 16:08:28 · history