curve #3236

y² + xy + y = x³ − x² − 430296049749789761073413883308424711159706580x + 1337861518400904123211779594356237371160319164544826734391001349047

a-invariants: [1, -1, 1, -430296049749789761073413883308424711159706580, 1337861518400904123211779594356237371160319164544826734391001349047]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 2206007226377691217596469738393439463302418193519281262196132127945323249850
discriminant (Δ): 4325739913502117867129958052519242202144451138654042228014013630458809616461490784870092521985849248681094348948099515721672359936000000
Faltings height: 24.7194
naive height: 319.9327
primes of bad reduction: 2, 3, 5, 11, 13, 17, 19, 29, 31, 97, 107, 193, 257, 313, 433, 461, 739, 883, 1259, 1601, 1741, 1873, 5563, 27997, 42437, 50131, 150131, 554951
regulator: 2102162548555379174577.5820846043226482113453199635976038764226369073129388
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:44:00 UTC
last updated: 2026-09-26 20:44:00 UTC

Witness: 18 independent points log in to add more points →

(873341970000860051749, 981189094633168587223120852082925)
(40976084134658725604389, 7246135315482990654209935387546605)
(-16238645765279695485675, -2010785771521054795196584427009459)
(186157749475626434264221/9, 30630131514051509241337847707151455/27)
(-21891963223217084862569/49, -424270030304775177193822692320192405/343)
(19131661320670698340869, 328881129964964528137837548063565)
(2282014423776644986149, 606468071280179215918647357177325)
(-268703181750331275531, -1205596962221235940220111463291635)
(22235290403974145546725, 1662351481790006522368357346311533)
(2670653688648746419269, 455782769080264504930749909387565)
(-19007695562649592551531, -1627716241666041460378231827921635)
(33367936973584629247119, 4912460924767910735506967614713565)
(-21954066498294463339611, -450739278772134641492262871413395)
(-1048519079571925373955819/49, -296533583052853272345509825387065205/343)
(62997732739415569840861/25, 64898451573140343591068169309457401/125)
(-317043130564298354002979/729, -24306039494028980739685529387681032025/19683)
(26787688505450614202085, 3005585023511496593979237417202093)
(779265679529900402837563941/10609, 20906230266920000324852067226367531083675/1092727)

Commentary

Voznyy (2024). Source: https://dujella.github.io/tors/z2old141516171819.html (curve entry 9). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:44:00 UTC · history