curve #3221

y² + xy = x³ + 13842139136873585658473535883368685x + 2310578587702287143012929957420435535344485058013217

a-invariants: [1, 0, 0, 13842139136873585658473535883368685, 2310578587702287143012929957420435535344485058013217]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 6098008034408866436674675605910737655832159836307310
discriminant (Δ): -2476092226426894410695170869028111842135118199889900449120195019566062513826231788992165893422560000000000
Faltings height: 18.9045
naive height: 250.0618
primes of bad reduction: 2, 3, 5, 7, 17, 23, 29, 31, 53, 67, 97, 131, 191, 223, 227, 313, 433, 2609, 3881, 8689
regulator: 6180116111881592.6507773320954856914800152486886638985640124714279697
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:43:17 UTC
last updated: 2026-09-26 20:43:17 UTC

Witness: 15 independent points log in to add more points →

(122297614621067374, 76371517653697877352456313)
(-83732000697566876, -23759198445595187766047687)
(25441067662683214, 51761028418949259675488953)
(73953816612765874, 61145100990660665734144813)
(34832022390612950254, 205575107269815996538774839193)
(40055275609337374, 54122961609888925621386313)
(41494881996289582, 54372819039684046901077753)
(-41972894336652626, -40689547714679739179783687)
(3136354086645978414, 5558520651891938736366377433)
(-82923898375968626, -24341709003753460397711687)
(-82532657984748626, -24616403601647205397451687)
(-5623744936084986, -47249931938065315398104247)
(62724008442088794946/529, 912025929019355834260539572171/12167)
(665259823510317866494/121, 17162848762624869194438013238523/1331)
(6421783216109867310766/10609, 526871501011269788489694652535551/1092727)

Commentary

Dujella (2009). Source: https://dujella.github.io/tors/z2old141516171819.html (curve entry 50). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:43:17 UTC · history