curve #205

y² = x³ + x² − 18031282130466232369801290576x + 937912694842853350198553539190914682552740

a-invariants: [0, 1, 0, -18031282130466232369801290576, 937912694842853350198553539190914682552740]
rank (lower bound): ≥ 13
torsion subgroup: trivial
conductor (N): 5883556108705604864069301892047856053057376997561594085263469332125323440
naive height: 206.8121
Faltings height: 15.0801
discriminant (Δ): -4824473724726869454399342069094860042611338540998910914446745354020291555955475532800
primes of bad reduction: 2, 5, 13, 17, 19, 23, 157, 241, 617, 41665714723, 46051797406879788477331351041654345220422732481
regulator: 296731964647705.6471367350377599179344630820568908069420258950288
submitted by: Seewoo Lee
submitted at: 2026-06-25 16:04:09 UTC
last updated: 2026-08-24 05:00:46 UTC

Witness: 13 independent points

(1019510314870079278/9, -1029407033233963549733458760/27)
(898379318706026386/81, -851447619821221046090639324/729)
(65554377206988649/144, -16124573198339556102902467/1728)
(89943481345135872/841, -11766652141274314098778430/24389)
(558225231481187845883/7360369, -1625706497883799859381407016320/19968681097)
(288028626599061/4, -908295957363279970007/8)
(36286970506582, -575667222416485993560)
(17527214855180285974742/58507201, 2119063633757101595805966005518120/447521580449)
(855715812309179, 24740771596287139505996)
(987441988220392, 30755977676121784722870)
(41176586031416048557327/31348801, 8313519254200782912639963937076910/175521936799)
(17435634703211710010126/200647225, -468872888856202121848899501476824/2842167942125)
(5731327856383289779122/383161, 433875811889536781966266940011540/237176659)

Commentary

Rank ≥12. Mestre–Fermigier construction from the integer 6-tuple a=[348, -600, -216, 492, 876, -900] with shift t=10852/13: 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 16:10:48 UTC · history