curve #6193

y² + xy = x³ − 17839658006712379880630872187355x + 29027077849798829875140020508779245542732275025

a-invariants: [1, 0, 0, -17839658006712379880630872187355, 29027077849798829875140020508779245542732275025]
c-invariants: [856303584322194234270281864993041, -25079395262226190296576354202876619554343483111161]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/3ℤ
conductor (N): 1188794000042104377137052333605459941590
discriminant (Δ): -628745676826672242232861042641158360789873578285369299764569666353232169737004862008832000000
Faltings height: 16.7258
naive height: 227.4923
primes of bad reduction: 2, 3, 5, 7, 11, 13, 19, 43, 53, 73, 107, 173, 659, 5323, 8599, 46831, 36841219
regulator: 353246022778.944313472042639989994899664852088931643015261315
submitted by: Andrii Prokhorenkov
submitted at: 2026-10-30 15:40:40 UTC
last updated: 2026-10-30 15:40:40 UTC

Witness: 11 independent points log in to add more points →

(-4802148192467898, 62890460942565550753881)
(2299836332676690, 12774465094900746098745)
(2386676677881510, 6680631345728813167545)
(2627319692000160, 17102000496347926883145)
(3821811178138110, 129110676857779630536945)
(61892965167915726/25, 741999261265801017273261/125)
(1872552815328462190/729, 241961763583461953349448435/19683)
(4058743837415298750/1369, 2373863354831252581494750765/50653)
(5116941984840865920/3481, 15881682160280915181893956005/205379)
(108305308803576937230/44521, 47281853381422215219433659675/9393931)
(182803023014813659176/73441, 131883758588297684071217989047/19902511)

Commentary

Specialization at T = 9095/7249 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 15:40:40 UTC · history