curve #6194

y² + xy = x³ − 3983617914448517595311825x + 3060380954899805772769306913365955625

a-invariants: [1, 0, 0, -3983617914448517595311825, 3060380954899805772769306913365955625]
c-invariants: [191213659893528844574967601, -2644169145033719008162521466415048111401]
rank (lower bound): ≥ 10
torsion subgroup: ℤ/3ℤ
conductor (N): 11484664266941817735462405540582990
discriminant (Δ): -202384434010796172350747665468852382714955388714799102819585828352000000
Faltings height: 12.7829
naive height: 181.5464
primes of bad reduction: 2, 3, 5, 13, 19, 61, 97, 101, 139, 353, 5394979, 753623680363
regulator: 130502129.6120305188307890935104175685147987762726356743609
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-30 15:40:45 UTC
last updated: 2026-10-30 15:40:45 UTC

Witness: 10 independent points log in to add more points →

(-1633343422700, 2282443796152497025)
(-1533731033990, 2358461387246441275)
(817037042460, 592477648843274025)
(1123303270090, 54455749379036635)
(1137030917650, 29706105259612675)
(1147650719200, 12338258788910125)
(1320384483250, 320080625988805675)
(14036889363082, 52085412509210006491)
(28721691700354/25, 1357926313533705023/125)
(1577756581634770/1369, 443361693168252439975/50653)

Commentary

Specialization at T = -5185/353 of the elliptic fibration of X648 given by: y^2 + (Lx+B)y = x^3, where L = 5098973881T^2 - 32101629380T + 51085836400, B = -2550648011199324171649113600T^5 - 231394683896910015292127210400T^4 + 1324630129545679262439545334000T^3 - 2330371426153230648960353010000T^2 + 1239686628515660725984584000000T.

last edited by Andrii Prokhorenkov at 2026-10-30 15:40:46 UTC · history