curve #4302

y² + xy = x³ + x² − 24659939083942618331155162x + 3293004701339712711619449610946548420

a-invariants: [1, 1, 0, -24659939083942618331155162, 3293004701339712711619449610946548420]
c-invariants: [1183677076029245679895447801, -2845156061966389360909423806457033693325]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 311560278166741566302026290678974420505803313066
discriminant (Δ): 955060695968585971930630599032768292666926589261520539877765012505723761842192
Faltings height: 13.6525
naive height: 187.0153
primes of bad reduction: 2, 3, 11, 13, 19, 31, 59, 67, 89, 97, 107, 163, 173, 467, 5209, 5563, 384751, 566173, 2031067
regulator: 26235877744144950362.7783975218781727429422582411984234644955105526
submitted by: The Wren
submitted at: 2026-10-16 19:11:26 UTC
last updated: 2026-10-16 19:11:26 UTC

Witness: 14 independent points log in to add more points →

(-233638418686479/64, 3421810908992302944589/512)
(-1625953519414176/529, 86077832541292499828798/12167)
(-2628308209303776/5929, 1716562337951822781382502/456533)
(-155343021591/256, 7449731751820672325069/4096)
(1255220076394241/256, -2089373320212366231271/4096)
(5020680424709, 2457798776455495427)
(-95905096388523485775060756/40462868047849, 1791129239134036695906771842625812434750/257386043555693546507)
(-52831127972117668829471/10610648064, 1775855153681368123805690509808255/1092981635776512)
(-9197154618016/25, 438663935488055845642/125)
(59071054726473516/25, -14356915140857711111671726/125)
(4902315627817905216/724201, 7460786935327696165935506854/616295051)
(-2851663900118905076/12852225, -136320792331660455678315779302/46075226625)
(3551231712810013216/29713401, 95471777198875641603363337354/161967748851)
(767962503566632745248749/95840395561, 530909954211835931612792063705173432/29670365498169941)

Commentary

Obtained by a rational 2-isogeny from my curve#4257, found by specializing the family associated with curve#802 at T = -334/279. This isogenous curve has the same rank and conductor, with smaller absolute minimal discriminant, Faltings height and naive height. The fourteen supplied points were transported from the source curve.

last edited by The Wren at 2026-10-16 19:11:26 UTC · history