curve #4953

y² = x³ − 67768923313753804827

a-invariants: [0, 0, 0, 0, -67768923313753804827]
c-invariants: [0, 58552349743083287370528]
rank (lower bound): ≥ 12
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 226796393437305875506046058539220172836
discriminant (Δ): -1984014849789551798926890920101098071969328
Faltings height: 6.7950
naive height: 104.8484
primes of bad reduction: 2, 3, 751, 443837, 7530145723
regulator: 72668711198990503.978593593817757236271354508190653030328413861
submitted by: The Wren
submitted at: 2026-10-20 23:43:48 UTC
last updated: 2026-10-20 23:43:48 UTC

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

(2509960585682710401/37699600, -3976494193290492754058844351/231475544000)
(2499697786817638401/29749350400, 3951904592425984541748582399/5131167956992000)
(2527598506376408433/42686039236, -4017827106319250127385607895/8819191822393016)
(2473751670088561323/68947731241, 3887901862773585964924897980/18104226321530539)
(2378830314069988593/162597658756, 3629060890694437422363847785/65564904330816904)
(2091080919036049107/352673262769, 2484113684398160755244141880/209439601847786647)
(2011840081833196401/395854888900, 1984730934655660155550845399/249060020449213000)
(2694099866929372401/203897402500, -4356578714015968313591187351/92069872098875000)
(2985924537477622401/384028090000, -4773218393510607042779406399/237982207373000000)
(3333488849233894401/553476481600, -5054899365151458364687398399/411764363251136000)
(8869251609450296979/1110715749025, 24593193931089720125409955308/1170588881476192625)
(2769778254588637565479/196748733691225, 143988507936737280104674538545442/2759738414097711075875)

Commentary

Obtained by a rational 3-isogeny from curve #4677, submitted by dujella: https://elliptic-rank.icarm.cloud/curve/4677. The source model is y^2 = x^3 + 2509960122731622401. Its public entry supplies twelve rational points and no commentary identifying an earlier source. Applied the degree-3 isogeny X = (x^3 + 4B)/x^2, Y = y*(x^3 - 8B)/x^3 with B = 2509960122731622401. The resulting model is Y^2 = X^3 - 67768923313753804827. Twelve supplied rational points transported through this map and checked exactly on the target curve. Rational torsion trivial. Same rank and conductor as the source; conductor 226796393437305875506046058539220172836. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 23:43:48 UTC · history