curve #341

y² + xy = x³ − x² − 3084587601676831625566042717x + 65997762542852427586433425766175664366441

a-invariants: [1, -1, 0, -3084587601676831625566042717, 65997762542852427586433425766175664366441]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 135728434608497948561915641226036813359924982522780290436813966763050
naive height: 201.5040
Faltings height: 14.5611
discriminant (Δ): -3336987098669501970890481504495636828867532662799128045348091937609448562993812500
primes of bad reduction: 2, 5, 7, 17, 23, 41, 73, 263884463, 133288999652357, 9421307797688753118486482507112040631
regulator: 4523809601657028110.312132294569182823380958526309006197033324151590846779793
submitted by: 7fff-zip
submitted at: 2026-08-24 19:19:04 UTC
last updated: 2026-08-24 19:19:04 UTC

Witness: 19 independent points

(5482409602445553, -405914927902343669302637)
(884323231909284, -26246969427969312049717)
(368135891521994, -6987253697844091612847)
(-26615414184979759/1849, 26057960202939920064602481/79507)
(-7970260217407031/2401, 32476377926270256180130172/117649)
(904980822794776875659/30184036, 3583817959818037571917260523827/165831093784)
(484656041357598914309/15452761, 624155922669691835316174109573/60744803491)
(51170778122056719450/1545049, 24782733151924756199323457503/1920495907)
(34685585230304, 27146086832673595923)
(65171193492693, 376526333758412072623)
(127918358983491134591/1885129, 1064456212683213642019927977116/2588282117)
(1568312459328623626/7569, 1899710732084472584519308819/658503)
(34804647475960, 28299409282181228181)
(2507912851611679, 125563393822782356826773)
(56840290468211327130719/1100401, -13551400570909999219173307035099028/1154320649)
(-2259441977470839766/83521, 8691047799105577021103678337/24137569)
(-29532855865420267/7921, 196187424194633242162992092/704969)
(2055990865266521/169, 382255036322074603474476/2197)
(26599240767639380/841, 213825979468701051047991/24389)

Commentary

New in this search: Mestre quartic family with roots [0,115,206,234,731,784], T=5494/7. 19 independent rational points certified exactly by Cremona-Brumer quadratic characters; quartic point box [10000000,10000].

last edited by 7fff-zip at 2026-08-24 19:19:04 UTC · history