curve #3214

y² + xy + y = x³ + 5951426409500892217936031442603949x + 168382775057670806272015446735529656531537361718498

a-invariants: [1, 0, 1, 5951426409500892217936031442603949, 168382775057670806272015446735529656531537361718498]
rank (lower bound): ≥ 15
torsion subgroup: ℤ/2ℤ
conductor (N): 1101143210509046379320062209869918451783745022537012850
discriminant (Δ): -25739361889740122269732465499891654536957110266298190776978099578261161073075859523788323390205050062500
Faltings height: 18.5292
naive height: 244.9204
primes of bad reduction: 2, 3, 5, 7, 11, 23, 79, 89, 97, 191, 241, 337, 347, 383, 1223, 1579, 2543, 4271, 12619, 31219
regulator: 2521992134207406.0938002433480880054620411317158434440512838388232684
submitted by: Natanael Wildner Fraga
submitted at: 2026-09-26 20:42:59 UTC
last updated: 2026-09-26 20:42:59 UTC

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

(-12424791682630248, -9618703592421505339970714)
(649064467008298733772/5041, -19828579149516242400501194545849/357911)
(-13817377740325239708/841, 198388696185516223761294953909/24389)
(267406725077795883/4, 235289186437254982731428663/8)
(7680740308109173692, -21287595448562703322767645134)
(33177298623841114783167/22201, -6051323485016661599273862091339666/3307949)
(4688944416563397926493/54289, 460739875476188684038560118520672/12649337)
(3476158666388532782148/11449, 212059593079260656203889309661923/1225043)
(20918078391406449524457/151321, 3547831924489188990973401942779054/58863869)
(42731990257498192073103/344569, 10729050986786556651175311836767418/202262003)
(-3534030474178283434347/146689, 186482162185567453519589074939382/56181887)
(180314539432342509037818/49729, 76585254534124261290293631017874172/11089567)
(-11296619573965647, -9985498106090620778112026)
(162430145824668823775955423/222695929, 2082134835367473153917387197736713666402/3323291348467)
(-607297056212820021753/28561, -27399578274434621895741420849626/4826809)

Commentary

Dujella (2009). Source: https://dujella.github.io/tors/z2old141516171819.html (curve entry 39). Published independent points; exact point coordinates checked.

last edited by Natanael Wildner Fraga at 2026-09-26 20:42:59 UTC · history