curve #3959

y² + xy + y = x³ − x² − 73710721598354467173070187x + 242486431596879408601509687290787809211

a-invariants: [1, -1, 1, -73710721598354467173070187, 242486431596879408601509687290787809211]
c-invariants: [3538114636721014424307368961, -209508276899687887515839125254331283998209]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
conductor (N): 2108987770780492075994930190
discriminant (Δ): 229921179523637184549007573553619706800136331315966054011987378837002332160000
Faltings height: 13.6898
naive height: 190.3002
primes of bad reduction: 2, 3, 5, 11, 31, 37, 67, 101, 137, 181, 191, 199, 331, 929, 947
regulator: 4885363903229.2747136166046105111272682457864311694270507
submitted by: Odbayar Shinebayar
submitted at: 2026-10-05 04:34:35 UTC
last updated: 2026-10-05 04:34:35 UTC

Witness: 9 independent points log in to add more points →

(90418383666522621131/15594601, 194970734236873131242288058666/61583079349)
(175142157432123014219/84364225, 7686948691091154773019926825802/774885406625)
(85095988299911205/21904, -12454067624049618816779881/3241792)
(93700961242952338368304451/34042685521321, 1544353637683157006074579466674849092654/198625827412240241131)
(47717120459388395/2401, -9577487157648958949295222/117649)
(2010145216694137281460395/357842043601, 510259450193681715704728808983869478/214060752640074599)
(24999274130820227682371/6969745225, 2865048099344333326031386458900114/581869180109125)
(627074160325964688024395/26331877441, 467929098575539953314698617788608182/4272900084228511)
(-974406782620689839735867461/2988474000961969, -2666926348912612301755247254645622198596998/163370719936725898718953)

Commentary

Unlisted rational-isogeny neighbor of curve#3674 (source contributor: Michael Rubinstein). Source equation and witness provenance are credited to that entry. Exact rank 9: 9 independent points certified by the unchanged ICARM verifier, with a matching unconditional upper bound computed using PARI/GP 2.17.2 ellrank. This is an additional representative of a known isogeny class, not a claim of a new world rank record.

last edited by Odbayar Shinebayar at 2026-10-05 04:34:35 UTC · history