curve #4923

y² = x³ − 1107675395524067283

a-invariants: [0, 0, 0, 0, -1107675395524067283]
c-invariants: [0, 957031541732794132512]
rank (lower bound): ≥ 12
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 60589865770340686157128399363506276 ☆ record for Mordell curves, rank ≥ 12
discriminant (Δ): -530040145758940322502559237631952902448
Faltings height: 6.1094
naive height: 96.6207
primes of bad reduction: 2, 3, 25969, 164357, 9611813
regulator: 13836989833967245.338897297773386611864750129473908840620785240
submitted by: The Wren
submitted at: 2026-10-20 22:56:34 UTC
last updated: 2026-10-20 22:56:34 UTC

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

(123108877178208543/118884971209, 2147450660000624830400370/40991181417949573)
(30782835583129803/29710382689, 346244602550944871914200/5121089532954863)
(31064016080570779/29164100625, 1581024433704070519564208/4980499284234375)
(32637794522594367/26005110121, 3909812152259471639052510/4193610063222581)
(130919613739938363/103257467569, 32007323155280523612192360/33180444856219753)
(32903467707711529/25453011600, 4166187102515421128072917/4060773470664000)
(35457844971513529/19788048900, 5999752989700423825491917/2783584838763000)
(39442313740159657/8555510016, 7788875944505145545935765/791350454439936)
(40499866018553779/4100481225, 8145741018018381083275708/262574315242875)
(163855249393035423/3913378249, 66326527755044413124683950/244809203122693)
(164003374977208843/2106534609, 66416910282873909133035580/96683618949273)
(164440053912792991/4871342025, -66681469585831145792589586/339995316634875)

Commentary

Obtained by a rational 3-isogeny from curve #4753, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/4753. The source commentary attributes the original Mordell curve y^2 = x^3 + 41025014649039529 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 = 41025014649039529, giving y^2 = x^3 - 1107675395524067283. The coordinate map is X = (x^3+4*B)/x^2, Y = y*(x^3-8*B)/x^3. 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 60589865770340686157128399363506276. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 22:56:34 UTC · history