curve #4921

y² = x³ + 334744316373626965368

a-invariants: [0, 0, 0, 0, 334744316373626965368]
c-invariants: [0, -289219089346813698077952]
rank (lower bound): ≥ 13
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 4150139160905439335634474289855580421312
discriminant (Δ): -48407223172801044410840508116875490034183168
Faltings height: 7.0612
naive height: 108.0429
primes of bad reduction: 2, 3, 174900953, 8860684741
regulator: 6466793116832014257.394781788733204745296872147069469108037305346
submitted by: The Wren
submitted at: 2026-10-20 22:55:04 UTC
last updated: 2026-10-20 22:55:04 UTC

Witness: 13 independent points log in to add more points →

(-36430494400452669623/5574401274289, 98153482505037209862074029655/13161256173417991913)
(-7173187810203168938/1896809299009, 43764938198562045109556678140/2612374916632248223)
(-26893655107378045127/8016650414161, 391165077033802161600605020495/22698095466492616409)
(-1780024149363490634/3043262805025, 97103737587578732996441246164/5308956747052087375)
(9091852464115604089/15101190300625, 1074026255860197859982376321937/58683603037986265625)
(4976057525469422866/4225840819225, 159324616945094250848976410336/8686997584468544125)
(33398336136361619737/19026145438609, 1530606712046936716778104996865/82990086710232281273)
(21793601746457051218/6636296362201, 328915299867337180466065007920/17095769694962358301)
(67187274167873518198/11655607668961, 912876476835337030800221678900/39792605905670591791)
(109505189928830977366/15487806057025, 1598976353443018553001054204164/60951563786149321375)
(544183359761328058657/70644983173249, 16708454638357186811494657983205/593775110335198720193)
(199694467582549453078/22404233022721, 3424596144063181062036989960420/106046202613008559231)
(1075404073155201721366/66631548724225, 36643213778619626795886289368164/543901005399334693375)

Commentary

Obtained by a rational 3-isogeny from curve #4755, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/4755. The source commentary attributes the original Mordell curve y^2 = x^3 - 12397937643467665384 to Noam D. Elkies, “j=0, rank 15; also 3-rank 6 and 7 in real and imaginary quadratic fields”, NMBRTHRY post, 30 December 2009: https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;6a3fad67.0912&S=. Applied the standard degree-3 isogeny from y^2 = x^3 + B to y^2 = x^3 - 27*B. Here B = -12397937643467665384, giving y^2 = x^3 + 334744316373626965368. The coordinate map is X = (x^3+4*B)/x^2, Y = y*(x^3-8*B)/x^3. Thirteen 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 4150139160905439335634474289855580421312. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 22:55:04 UTC · history