curve #4730

y² = x³ − 43533050058824997522916x

a-invariants: [0, 0, 0, -43533050058824997522916, 0]
c-invariants: [2089586402823599881099968, 0]
rank (lower bound): ≥ 6
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 696528800941199960366656
discriminant (Δ): 5280040608225233465542032213204339387686129426586004579587920529362944
Faltings height: 12.0680
naive height: 167.9970
primes of bad reduction: 2, 17, 41, 103, 641, 2267
regulator: 28669540.740131483585386386093918885123217944076883
submitted by: David Renshaw
submitted at: 2026-10-20 21:26:34 UTC
last updated: 2026-10-20 21:26:34 UTC

Witness: 6 independent points log in to add more points →

(-4455350155147/49, 19422836135704192827/343)
(664776050835949/81, 17134548249797036222255/729)
(433529065593702/121, 9011346481061668014336/1331)
(73932354598417/324, 257387620265847904679/5832)
(126918167055713017/125316, 44245436258381779900899091/44361864)
(24191462379842870240773569009457/104760622353480814224, 50986618599036294312283435416075741312135834409/1072252591811650831161964813632)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 208645752554. Source: Watkins et al. electronic data http://magma.maths.usyd.edu.au/~watkins/PTS.r6r7 (paper Sec. 15); Watkins, Donnelly, Elkies, Fisher, Granville, Rogers, 'Ranks of quadratic twists of elliptic curves', Publ. Math. Besancon 2014/2, 63-98, doi:10.5802/pmb.9 (https://pmb.centre-mersenne.org/item/10.5802/pmb.9.pdf).

last edited by David Renshaw at 2026-10-20 21:26:34 UTC · history