curve #342

y² + xy + y = x³ + x² − 1391553064700695297387537864163x + 632237601708869371752246880018016126067921281

a-invariants: [1, 1, 1, -1391553064700695297387537864163, 632237601708869371752246880018016126067921281]
c-invariants: [66794547105633374274601817479825, -546253287876463638153044596585872992436315085625]
rank (lower bound): ≥ 19
torsion subgroup: trivial
conductor (N): 25582449424667650913572690219204060112186535917088135557790710631313352389050
discriminant (Δ): -224544518159358460458242318645476296228674299566935193705566332442671710143907131904000000
Faltings height: 16.0778
naive height: 219.8388
primes of bad reduction: 2, 3, 5, 7, 17, 10982045197, 130503054599564132532868602134756795265311693363089881885691589
regulator: 127836638655370553024.8093793090760805015496216256853242770719472002102863646
submitted by: Andrii Prokhorenkov
submitted at: 2026-08-24 19:19:09 UTC
last updated: 2026-08-24 19:19:09 UTC

Witness: 19 independent points log in to add more points →

(222580170941029009509963915/1216609, -3320700499005602098592328210951318215686/1341919727)
(15997794013166186205/1681, -63516948951917242882076286538/68921)
(24523399592876245268555/4116841, -3770151461228919948964463677342042/8353070389)
(63608988183595951879/18225, -481451301097855542258628700278/2460375)
(2412659329425315, -106389865484273319795458)
(29935033694463679/225, 71551113611155031077107062/3375)
(699522691224164861327035/2296038889, 1692150122388151296454402555985151446/110019295444213)
(629988782486369715/961, 39119259381717047125170322/29791)
(357793325662813235/529, 8219560419083748633332914/12167)
(5136391303343585155/6561, 2528720778321622076503960702/531441)
(931443977980185, 12008054729918060563432)
(4636434350273393695/1089, 9634954864148332440193362614/35937)
(4647240686381935, 307459095765403595866682)
(80483979169908302084449415/1351738756, -721904184596216915316984299158925280143/49698027103096)
(37629837485889561389035/56385081, 376744110291641780530902051184678/423395573229)
(1349983754625485038489/151321, 49188018416713726738684244643472/58863869)
(19313638877029794697035/4044121, 2608940737441192802891687337307642/8132727331)
(8476247175871114835/361, -24647181211649950105309929062/6859)
(9171897417206983035/11881, 5502037771750027411464464278/1295029)

Commentary

New in this search: Mestre quartic family with roots [0,115,206,234,731,784], T=9693/14. 19 independent rational points certified exactly by Cremona-Brumer quadratic characters; quartic point box [10000000,10000].

last edited by Andrii Prokhorenkov at 2026-08-24 19:19:09 UTC · history