curve #3609

y² = x³ + x² − 326317355466512162296636500x + 2162588636175573089988210707820353700000

a-invariants: [0, 1, 0, -326317355466512162296636500, 2162588636175573089988210707820353700000]
rank (lower bound): ≥ 18
torsion subgroup: trivial
conductor (N): 69920387819660816126042026646958403496457502354638120
discriminant (Δ): 203451294497818353546774978785472325873481688033784953253734536788817109079200000
Faltings height: 14.1666
naive height: 194.7633
primes of bad reduction: 2, 3, 5, 11, 13, 19, 41, 43, 53, 83, 97, 124633, 120383865861207202704904688575001
regulator: 339630882848320791620.55020667436182611174225135204122566724794105424825212
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:18:37 UTC
last updated: 2026-09-29 19:18:37 UTC

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

(-12449942428650, 65539809056701272150)
(33421391609880, 169079712693124456080)
(265266321704840, 4310613681978930192360)
(41747486030347800, 8529929690631818858017200)
(-25384872461008975991475/2006860804, 5872254898510241448278768283789675/89903350297592)
(-1707664217080516767825/828461089, 1267736478968080443439409597726850/23845595524687)
(-150917693160043409640/11621281, 2570580763088509924614479863440/39616946929)
(408911916859100/49, 1556456625126348877800/343)
(13238843074278780/361, 1363479978316852767107880/6859)
(27096545514658710/2209, 243748926061613753193270/103823)
(237955592008571610/37249, 132305296003158118553971830/7189057)
(256050045710553288/52441, 314466738595663017498787224/12008989)
(38597866574521809990/2920681, 62780121303071566896044443890/4991443829)
(100835234241683867175/885481, 1000484315718887700735327171450/833237621)
(359086212212746841085/25260676, 2527821071054457690951069801885/126960157576)
(382849791711193291800/7219969, 7100517940874936236084062639600/19400056703)
(1357371718079790354510/93180409, 20117735406504191654065694982870/899470488077)
(1002158373652278405775770/147814874089, 919620946241976653060460186219654690/56829941196375563)

Commentary

Specialization at T = -7/20 of the rank-17 fibration of X1008, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #751 (same T), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:18:37 UTC · history