curve #3230

y² + xy = x³ − 15084248699780772863935958932958691919487759038x + 712998263524553933194323224272640402753488391782195083964735452378692

a-invariants: [1, 0, 0, -15084248699780772863935958932958691919487759038, 712998263524553933194323224272640402753488391782195083964735452378692]
rank (lower bound): ≥ 18
torsion subgroup: ℤ/2ℤ
conductor (N): 11148693462601817177654682417706894412080396884801101938270196382625118197171415850
discriminant (Δ): 45685613684283730119718108084836570398355636824798157903054807907216734855956263930098210229919953180569796832888069404795464646656000000
Faltings height: 25.2472
naive height: 330.6035
primes of bad reduction: 2, 3, 5, 7, 23, 37, 101, 109, 149, 157, 179, 181, 223, 251, 311, 449, 479, 769, 953, 1609, 2017, 2221, 2239, 2857, 2897, 29363, 31121, 100799, 154667, 286397
regulator: 16861403182817324768304.658061216998010879418411363426347847617106410524859
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:43:43 UTC
last updated: 2026-09-26 20:43:43 UTC

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

(68790207655603411764572, 933434788606925260277495740318514)
(75146818077639206122268, 1955090501051966617107495670220402)
(71552178910534363959836, 118810273295578069452033898236530)
(79800174118214287627052, 4176782161160112807869494478045474)
(-26056625866204193392228, -32990177338448883370413673896591886)
(57822733371906922185272, 5840741403435611920296626522574614)
(46930402575736468274972, 10413974265337900102865233550528114)
(-1271412283523693576595716/9, -134140753111277735447168562497402362/27)
(71577645911288207847452, 145793855395096034170020490415474)
(-80040057074407483958344516/1369, -1891916031284604976258850678740365806998/50653)
(110120016426075878074652, 19679479165460978760227805583132274)
(26181801390794993273786492/361, 4782345477531878202967108940414849126/6859)
(1395508116955203707018390356/3481, 49926688837204336670502912175839738092270/205379)
(69214241858097938020652, 729351269806999870661158587057074)
(11837072424848177728759288/169, 641065220897527508798007288064028878/2197)
(3445552394613504550097468/49, 3125038929087975249545667774770462/343)
(128724695954406137506765532/1681, 181567994172562764032885249291630909794/68921)
(3530946993452506014985448/49, 156909097600250596650752037262744382/343)

Commentary

Voznyy (2023). Source: https://dujella.github.io/tors/z2old141516171819.html (curve entry 6). Published independent points; exact point coordinates checked.

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