curve #3181

y² + xy = x³ − 73262771788012628080963454016709537315x + 240598790801630018163569184912325717758514948008409652225

a-invariants: [1, 0, 0, -73262771788012628080963454016709537315, 240598790801630018163569184912325717758514948008409652225]
rank (lower bound): ≥ 13
torsion subgroup: ℤ/3ℤ
conductor (N): 52729481509147959808917342178856082795095009006395410
discriminant (Δ): 159396482572821488761152834672281752095001214878216681381644951287662698319959362058601852715817241340895744000000
Faltings height: 20.5820
naive height: 273.1750
primes of bad reduction: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 61, 67, 1723, 21013, 140977, 4562524327761901
regulator: 147342407296.6840588388537486909403919882671020236611767871245115
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:41:11 UTC
last updated: 2026-09-26 20:41:11 UTC

Witness: 13 independent points log in to add more points →

(4457502222769039370, 1611641339515815829274369045)
(4198800350419611170, 2647181824163160663752575085)
(4663784697212755040, 598710342059502067618068575)
(4600449196285678970, 960025506026782647465074945)
(4554776706911186990, 1181769967892808613450633385)
(2380239538380692222, 8927552162407067648923077497)
(23702110592754993710, 108718564570356475210335730985)
(4305435373222259510, 2231488968639382184715604505)
(4618909929490933310, 863707409335431200909470985)
(4372416906624676220, 1963518441135919270046813945)
(36249530734715794730, 212644711673400126644106584885)
(4693924773486814190, 360230832576196960473000425)
(4450103380274798480, 1642908737129617992291509135)

Commentary

Eroshkin (2007). Source: https://dujella.github.io/tors/z3old891011121314.html (curve entry 4). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:41:11 UTC · history