curve #4669

y² + xy = x³ − x² − 11049384374804165994x + 14142702381810938408153672208

a-invariants: [1, -1, 0, -11049384374804165994, 14142702381810938408153672208]
c-invariants: [530370449990599967721, -12219294855497983759687072932981]
rank (lower bound): ≥ 9
torsion subgroup: ℤ/3ℤ
conductor (N): 28770513284662855740810
discriminant (Δ): -70468927605301820107969942746781526333517727784179687500
Faltings height: 9.6718
naive height: 143.1611
primes of bad reduction: 2, 3, 5, 19, 29, 37, 41, 61, 101, 157, 487, 499, 1627
regulator: 397465713.58890375478916996958116074927082912436706743452
submitted by: jesper-petersen
submitted at: 2026-10-20 19:31:14 UTC
last updated: 2026-10-20 19:31:14 UTC

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

(9206559951473883/4853209, 31307944407363051847428/10691619427)
(1691313581417418/877969, 2026156416393671640792/822656953)
(3408469183998/2209, 2875617980735103972/103823)
(52727974679007/27556, 11156044445260773669/4574296)
(2624668788393/1369, 121831377791656617/50653)
(55738191363265902/14280841, 9420723679374010231812546/53967298139)
(960322626118983503488/477794765529, 2425265894690232442309819837612/330264642392314083)
(-15883280681628498/17048641, 10820592349766961549333996/70393838689)
(7686643382714196928/3079473049, 7857439205748301205440874748/170889197908157)

Commentary

Exact rank 9 established by Magma 2-descent. Independence also checked using a 10x10 mod-3 reduction matrix including the rational 3-torsion point, with determinant 2 mod 3. Found by a precomputed Mestre–Nagao sieve of the rank-5 C3 K3 deformation, specializing w=7/4, t=81/64.

last edited by jesper-petersen at 2026-10-20 19:31:14 UTC · history