curve #5665

y² + xy = x³ − x² − 49417220995586825961480136196599x + 134161999624082287145388308818220459371558368769

a-invariants: [1, -1, 0, -49417220995586825961480136196599, 134161999624082287145388308818220459371558368769]
c-invariants: [2372026607788167646151046537436761, -115915967675207085419495763772188069217317012151005]
rank (lower bound): ≥ 23
torsion subgroup: trivial
conductor (N): 39766247330849852625400691919003958147425632867140192142342159145143318089727618106256358
discriminant (Δ): -52245134340754521478367181447338914237836681014645987277761982737844162166192683033708240678548
Faltings height: 17.0325
naive height: 230.5539
primes of bad reduction: 2, 13, 23, 62219, 80713, 105269, 81143401155738689129586097, 1550219530537495813826821888244255452594796951
regulator: 3948132352215375837475295.18574242589674939344904111129988468820155636982342401943976
submitted by: apkorte
submitted at: 2026-10-24 01:50:41 UTC
last updated: 2026-10-24 01:50:41 UTC

Witness: 23 independent points log in to add more points →

(2985932527785413, -115010733735269963763483)
(3925433208024677, -25789556742181017321002)
(15428007681612430/9, -6302473697730760028476711/27)
(3305485486472648, -83250237364382725302015)
(-182832429369900451/25, 40392242980054293156401022/125)
(-8102125461937392, -51836852050317855153983)
(982074627174144511/81, 836699916632910209929120670/729)
(-10736741490716899237/1444, -16509953470258755704146050069/54872)
(12476377210796558, 1208174997852165389733267)
(-31823455420499949/25, -55199133140981433527606389/125)
(1120429478715454, -283196104521004379947657)
(1958845946320625, 211843148993107542971113)
(4093923564926700, 21603085362691824690713)
(7147674171821070, -382247423497391872973677)
(16669610099191223/4, -195443103810851588246781/8)
(2255031181895568, 184910335036104232897967)
(4783217912989033, -84998449205504904559983)
(7354281973758510, -410479686751668982739953)
(3878099938804206, 29023765522676838633955)
(415180525239935150/289, -1263449723320242501777913869/4913)
(-12061009200263005/4, 4045766032384054639054961/8)
(3798862128343104, -35433213079163728882687)
(36519098806671732828/7921, -46371466085001036399942543965/704969)

Commentary

Specialization at T = -53/359 of the rank-17 fibration of X1008 given in curve#751. Found by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.

last edited by apkorte at 2026-10-24 01:50:41 UTC · history