curve #5057

y² = x³ − 1216065811286574592404154911960025x

a-invariants: [0, 0, 0, -1216065811286574592404154911960025, 0]
c-invariants: [58371158941755580435399435774081200, 0]
rank (lower bound): ≥ 7
torsion subgroup: ℤ/2ℤ × ℤ/2ℤ
families: y² = x³ + ax (quartic twists); y² = x³ − n²x (congruent-number curves)
conductor (N): 38914105961170386956932957182720800
discriminant (Δ): 115093609507936919409641053742415529011972910633773706133981948113516039062682310893225893115201000000
Faltings height: 18.0813
naive height: 240.1564
primes of bad reduction: 2, 5, 17, 19, 23, 97, 101, 5857, 16361
regulator: 15918138125.45086627340207074840783990989792874748850
submitted by: David Renshaw
submitted at: 2026-10-21 01:50:34 UTC
last updated: 2026-10-21 01:50:34 UTC

Witness: 7 independent points log in to add more points →

(-4722990064421775862130820/343694521, 23939471560987507426068193596845052150/6371752724819)
(2467669623793536742981921/28601104, 3545680100603658249099354354497724081/152958704192)
(3671870380630485299038793787761/28433529936100, 6774694649450316290634485093304726949164662241/151616396013565391000)
(-6420836488965006020/20331081, 1796531412254699582287070437921850/91672844229)
(-1000948660184917398285646663680/737889578359129, 25724502872104818407018516038834945456953076800/20044123266965518328867)
(24856833666144819741201/712336, 141274038084225440328532814139399/601211584)
(427892760906119427997965080177854973820/8344855487796500830249, 6489122984768962278367243549095461743978310452263572831850/762304055532282660916046557268693)

Commentary

Congruent-number curve y^2 = x^3 - n^2 x with n = 34872135169596005, rank 7 example from Watkins, Donnelly, Elkies, Fisher, Granville, Rogers, 'Ranks of quadratic twists of elliptic curves', Publ. Math. Besancon 2014, doi:10.5802/pmb.9; points from https://magma.maths.usyd.edu.au/~watkins/PTS.r6r7.

last edited by David Renshaw at 2026-10-21 01:50:35 UTC · history