curve #5498

y² + y = x³ + 69309776556

a-invariants: [0, 0, 1, 0, 69309776556]
c-invariants: [0, -59883646944600]
rank (lower bound): ≥ 5
torsion subgroup: ℤ/3ℤ
family: y² = x³ + k (Mordell curves)
conductor (N): 7485455868075
discriminant (Δ): -2075261094551794511266875
Faltings height: 3.3449
naive height: 63.4468
primes of bad reduction: 3, 5, 31, 43, 79
regulator: 138173402.858430041133341496886647993445732904087
submitted by: The Wren
submitted at: 2026-10-22 12:45:17 UTC
last updated: 2026-10-22 12:45:17 UTC

Witness: 5 independent points log in to add more points →

(2673476711435825641/677345284636164, 6375536685973841378214014783/17628492194910385928712)
(1471102376030870360804281/4190897221468937604, 1784286921997433042323692406161552775/8579471093788049760861357192)
(-7474516509169964951/7839360641146896, 181586958065230459936695113771/694098276822291319930944)
(1672195204486820240209/451520465978831076, 105147097353873879233867616560885/303400407959265799811713176)
(-147849574968901954439/65618424107752356, 4043597890279532679193345336639/16808876807049960153660504)

Commentary

Rational 3-isogeny partner of hdeping’s ICARM curve #5109: https://elliptic-rank.icarm.cloud/curve/5109 The source model is y^2 = x^3 + x^2 - 1871363967019. Applying the standard rational 3-isogeny and the recorded rational scaling gives the target model [0,0,1,0,69309776556]. The five submitted points are exact transported points. PARI/GP 2.17.4 returned rank bounds [5,5], torsion Z/3Z, conductor 7485455868075, and minimal discriminant -2075261094551794511266875. Exact quadratic-character certification at 12 good primes gives an F2 matrix of rank 5, proving the five submitted points independent modulo torsion.

last edited by The Wren at 2026-10-22 13:01:25 UTC · history