curve #5712

y² + y = x³ + 664806410576848

a-invariants: [0, 0, 1, 0, 664806410576848]
c-invariants: [0, -574392738738396888]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 63643329150346501736733922108041
discriminant (Δ): -190929987451039505210201766324123
Faltings height: 4.8730
naive height: 81.7842
primes of bad reduction: 3, 53, 514867, 97450543
regulator: 17694714098328871.4685930978468990786967605394450949808376390
submitted by: jesper-petersen
submitted at: 2026-10-24 14:44:12 UTC
last updated: 2026-10-24 14:44:12 UTC

Witness: 11 independent points log in to add more points →

(-664453672580042/7625306329, 1184729648398231703909/665864624567267)
(-367602067989008/13430260321, 39506708513774912810783/1556419438340369)
(627991749450658/27430978129, 118193550875799636795944/4543200890659367)
(1097683550721532/32843950441, 157722632025331446178043/5952276294471989)
(1676630214933994/38956890625, 209805095534917915087097/7689116287109375)
(44107332706547272/263137194961, 9895536151030000609914263/134981223761949209)
(5740923125778673897/147937498657936, 48390549011662078083320007467/1799357286926534618816)
(32047566782568164092/1558389236120721, 1596554406946021566544231801328/61519657617976239192519)
(5006750861175341663809/133423319850264900, 1305579219758542819577280312220627/48735745516318424938893000)
(36103134693051312076/705337190494225, 529473746118182612800315854199/18732475592525868981625)
(-491199449993618726470310577251500183631/28801483171448682936254982426008100, 125557749306239311069567570741857725089538250461658744647653/4887899630234335876846185728423614965221222489271000)

Commentary

The rationally 3-isogenous partner of the Burkhardt-quartic Mordell curve obtained from parameters (u,v,w)=(-9/4,1/2,-3/2) and t=-617/824. Eleven rational points were reconstructed through the explicit 3-isogeny and transformed onto the minimal model. Their coordinates were verified using exact rational arithmetic, and PARI/GP 2.17.4 returned matching computational rank bounds [11,11].

last edited by jesper-petersen at 2026-10-24 14:44:12 UTC · history