curve #5723

y² + y = x³ − 272004068781569944

a-invariants: [0, 0, 1, 0, -272004068781569944]
c-invariants: [0, 235011515427276431400]
rank (lower bound): ≥ 12
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 1278481768134837687888432658099974675
discriminant (Δ): -31962044203370942197210816452499366875
Faltings height: 5.8754
naive height: 93.8123
primes of bad reduction: 3, 5, 19181, 756315294737
regulator: 78941393174497501.626398144799322853750915674246077390589655795
submitted by: jesper-petersen
submitted at: 2026-10-24 16:54:40 UTC
last updated: 2026-10-24 16:54:40 UTC

Witness: 12 independent points log in to add more points →

(23098001843145164281/1944870111396, 111000862908343899236863987201/2712288629171302056)
(1879999901150747815801/7348404110436, 81514850450374865160375587341559/19920009772145246184)
(362869680877515934/7056504009, -218587634270925858555453728/592767506268027)
(29721751031970156934/4429791555849, -161963051446014716263123021211/9323413296136281243)
(383835796629307576/159150729969, -235486762395189775594089265/63491114761642953)
(40057085211273430009/54677240092836, -140754287966828932832464547585/404305712205907075416)
(1334615051449197304/2040992248689, 254201474199445566147162127/2915828879221312137)
(7054638608161228066/7386692315409, 15539167842088924558078160912/20075899549357404423)
(82910648523373590184/39435297741009, 743815182871199270562516663272/247643929294994490777)
(5493189636179807158096/1618043786757225, 405716003838720271115909185142269/65085673788587501260875)
(16089769082832175868464/24702091103282121, 255732460267120032153736233853640/3882402684888745539345669)
(1753197306090620953337134/67290766734720321, -2321361900580334578411512565867951999/17455537052853333634969119)

Commentary

The rationally 3-isogenous partner of the Mordell curve obtained from the Bruin–Nasserden parametrization of the Burkhardt quartic using (u,v,w)=(3/5,16/5,-14/5) and t=1102/751. PARI/GP 2.17.4 returned matching computational rank bounds [12,12].

last edited by jesper-petersen at 2026-10-24 16:54:40 UTC · history