curve #5678

y² + y = x³ − 29520895752581767

a-invariants: [0, 0, 1, 0, -29520895752581767]
c-invariants: [0, 25506053930230646472]
rank (lower bound): ≥ 10
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 2832432250889733342553290483
discriminant (Δ): -376480779567033684957896798446207203
Faltings height: 5.5052
naive height: 89.3709
primes of bad reduction: 3, 7, 61, 23986672649
regulator: 87575557044820.40305273532473645919580545434394231804371052
submitted by: jesper-petersen
submitted at: 2026-10-24 07:36:45 UTC
last updated: 2026-10-24 07:36:45 UTC

Witness: 10 independent points log in to add more points →

(896314903801/2755600, 319970590931237549/4574296000)
(1170079555633/195364, 1265590617608197661/86350888)
(1175346310273/484, 1274232777657301821/10648)
(97265526278929/109411600, -938894776972298531033/1144445336000)
(2339466827161/6100900, -2469918815834805841/15069223000)
(422973797558929/1314787600, 2928642859247339983783/47674198376000)
(418539123697/948676, 219346276486559413/924010424)
(908575011397202857/2910149399056, -149884729876955115296122613/4964470422240015296)
(25718341550693521/25430680900, -4065150212278034760977819/4055430683123000)
(78992488433497/247307076, 215360581045228949087/3889151077176)

Commentary

Exact-rank-10 Mordell curve discovered in October 2026 using the Burkhardt quartic construction, with parameters (u,v,w)=(3,2,1) and specialization t=-221/770. Exact rank [10,10] independently verified with PARI/GP 2.17.4 using ten rational points checked exactly on the minimal model.

last edited by jesper-petersen at 2026-10-24 07:36:45 UTC · history