curve #744

y² + xy = x³ − 1520516072003547319024574245441524595x + 720815153586924394151608190001252327382142242741168481

a-invariants: [1, 0, 0, -1520516072003547319024574245441524595, 720815153586924394151608190001252327382142242741168481]
rank (lower bound): ≥ 22
torsion subgroup: trivial
conductor (N): 1361362064915177000972459768633252512530237534503251316734019645760857255743706022
naive height: 261.5499
Faltings height: 19.5752
discriminant (Δ): 528539827264313417385485714500388633927564851884176956989921760052400125799104625018634077682668074266263552
primes of bad reduction: 2, 3, 7, 11, 17, 23, 29, 59, 71, 163, 167, 2278990305852176197542295097626626443712221537052155162100626666591
regulator: 8205561413943454213804.241979962723574424991990899386792227237609110123851170521318
submitted by: Valery Asiryan
submitted at: 2026-09-16 18:16:52 UTC
last updated: 2026-09-16 18:16:52 UTC

Witness: 22 independent points log in to add more points →

(-5646163849602565729/4, -1870427763056769630403873667/8)
(-1337553221278447864, -601363931844141385678691689)
(-1213043223832155694, -883349009151119478588699859)
(-1111933852726638898, -1018202772524290544033719591)
(2149961010254713502957/21904, -2452895745232115405570168646834349/3241792)
(320620110186195710, -516009680577351530692833271)
(33466427046148811726/49, -10381512094593451226905155457/343)
(6189042147675291454/9, -535944951957823804749367069/27)
(731922764680219310, -3722084560116820063691239)
(839805271789504526, -190183163056310061427014247)
(942271629479883494, -353123485904700011495388559)
(1520220112654338854, -1386589307326212291832916815)
(1868371908749638798, -2098109551691371482856376807)
(2900847842262652238, -4551969542403383425377296167)
(7741480144459985702, -21281462847243430210783249231)
(10068684994629036974, -31720002670055142350442650599)
(26833236869801478734, -138854086914311188924572144679)
(681953645839846886, -32304269795931053711215447)
(731762190580787102, -46035518929386678245191)
(672135893788056808, -49702234944776340955233161)
(2343657942221190823118/1681, -78912847395034362669870480755471/68921)
(1391439632390604682622/1849, -4162639411505526696478481021989/79507)

Commentary

Specialization T=-41/94 of the family identified by wgxli in the commentary to curve #302. The points were obtained by specializing reconstructed generic sections and running minimized quartic searches. The 22 listed points are certified independent by exact quadratic-character certificates.

last edited by Valery Asiryan at 2026-09-16 18:16:52 UTC · history