curve #3653

y² + xy = x³ − 70125739284257541584773900448595x + 224743012466763550759810773071275962457173164961

a-invariants: [1, 0, 0, -70125739284257541584773900448595, 224743012466763550759810773071275962457173164961]
rank (lower bound): ≥ 16
torsion subgroup: ℤ/2ℤ
conductor (N): 2050943799731838770795008718676624626173063235165279006502
discriminant (Δ): 250437983262427361284766549940053429335781466528186256238429284746113038429474020987452918218752
Faltings height: 17.1410
naive height: 231.5971
primes of bad reduction: 2, 3, 11, 47, 137, 167, 179, 293, 307, 353, 433, 439, 1193, 1697, 2647, 4787, 5393, 10211, 20161, 26863, 34961
regulator: 15324107086174751135.10100142294253329388275312204698776390111076470537
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:27:27 UTC
last updated: 2026-09-29 19:27:27 UTC

Witness: 16 independent points log in to add more points →

(4011125375232078, 89416913858436189809529)
(6576372845569758, 219064937612195671731273)
(20693955204040404, 2763247892048250754871955)
(101814396335978814, 32380710357493942041363225)
(1768348194780203985/64, 2254042173260881531239849213/512)
(6661029569181270834/25, 17183040655816079582336464713/125)
(7140827896366670526/1369, 1500999993494766220087386165/50653)
(14304799441228871058/1681, 34060039425423379190003026329/68921)
(14678575361306770758/3481, 13018372123599098657811079971/205379)
(18076771028524845598/81, 76803271743698979341113018817/729)
(322330764564225228540/9409, 5628086295050009828955380074473/912673)
(164548871336383594449456/28058209, 18315428816338941065096354853878223/148624333073)
(516542169431021749047642/114340249, 14541921904075744266251065549375761/1222640282557)
(3416375559827424969684606/759829225, 384845062875762659866079899322823501/20944692587125)
(20377855121971249896247182/2999424289, 40871033643489795133236697377701434839/164269470035663)
(598946995332182445750772854/112766341249, 1742280565256025560591922699365082985623/37867726755802943)

Commentary

Specialization at T = -971/167 of the rank-9 fibration of X400, y^2 + (Lx+B)y = x(x+D)(x+E), given in the commentary of curve #802 (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 19:27:27 UTC · history