curve #3623

y² + xy + y = x³ − 8935444839098988438628x + 325066122267677796797923301076998

a-invariants: [1, 0, 1, -8935444839098988438628, 325066122267677796797923301076998]
rank (lower bound): ≥ 11
torsion subgroup: ℤ/2ℤ
conductor (N): 498653576436666580377542225670
discriminant (Δ): 10653111597901377068503021060982585834191204181595096350404687500
Faltings height: 11.3044
naive height: 163.2465
primes of bad reduction: 2, 3, 5, 7, 11, 17, 19, 23, 41, 89, 113, 167, 227, 503, 631, 2063
regulator: 1183232603745.82565082748542605328972091641395828623337684866
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:21:31 UTC
last updated: 2026-09-29 19:21:31 UTC

Witness: 11 independent points log in to add more points →

(90849893599, 16221235778859725)
(-13297486649964311/175561, 1752102709239385668748375/73560059)
(1384678995481/25, 33180825114705781/125)
(5636785329979/121, 4189010028203017225/1331)
(26050808170561/289, 77999130572462486725/4913)
(272681396989564/5041, 3823877038545377455/357911)
(630646071597559/7225, 8902764273781351788953/614125)
(25057580156841469/712336, 4429196406502496956099295/601211584)
(21212032149651664/72361, 2945345560626496493355850/19465109)
(69398666062871364/185761, 17746017088081257746357375/80062991)
(251739970325161208194/439279681, 3942967583352781096634701345625/9206862834079)

Commentary

Specialization at T = -106/5 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 19:21:31 UTC · history