curve #197

y² + xy + y = x³ − x² − 15848001975027331832x + 24771530688804644612096284539

a-invariants: [1, -1, 1, -15848001975027331832, 24771530688804644612096284539]
rank (lower bound): ≥ 14
conductor (N): 158660341771519260190542491906863061372490
naive height: 144.2821
Faltings height: 9.9187
discriminant (Δ): -10343870453933375939366089310754943399947700700823152640000
primes of bad reduction: 2, 3, 5, 7, 17, 131, 6811583, 117316757, 141514035854442770479
regulator: 91751195039.7385955277149593413022755814531860544813979568131122369
submitted by: Seewoo Lee
last updated: 2026-07-01 22:43:40

Witness: 14 independent points

(-64560819529/25, 27521684792689179/125)
(36123117509/25, 8740159303518117/125)
(12682101068781/4225, 17895139115611159401/274625)
(272865715950557/167281, 3906800984102749854129/68417929)
(53671483231/9, 10186605986437651/27)
(428560504493/49, 255573047630702943/343)
(12449957567, 1325597777680281)
(15851815757, 1938237977452041)
(897117523463477/44521, 26382156112824679385091/9393931)
(7586604431813/169, 20816879163190357437/2197)
(-1901953423, 219166447716621)
(-313569788467/289, 991035603369385113/4913)
(-2964269157/4, 1520192145743473/8)
(2016208521369317/885481, 18467414160521026345821/833237621)

Commentary

Rank ≥14. Mestre–Fermigier construction from the integer 6-tuple a=[-138, 138, 162, -60, -90, -12] with shift t=1227/5: 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 14 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 16:10:10 · history