curve #4926

y² = x³ + 26968145165498169

a-invariants: [0, 0, 0, 0, 26968145165498169]
c-invariants: [0, -23300477422990418016]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 35915103884809000742847108264324 ☆ record for Mordell curves, rank ≥ 11
discriminant (Δ): -314185328784309138498426503096306352
Faltings height: 5.4902
naive height: 89.1900
primes of bad reduction: 2, 3, 998820191314747
regulator: 1800389316206584.13810950663803342143097521190098316685228960
submitted by: The Wren
submitted at: 2026-10-20 22:59:15 UTC
last updated: 2026-10-20 22:59:15 UTC

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

(-2952350679769457/10284190921, 59996187505479105950696/1042930085489531)
(-100859668763297/24753043561, 639540280525915508039596/3894421096495691)
(1335754957762383/30516945481, 876821776297946290542096/5331035723021371)
(4421059874056543/41376034921, 1413044908770171545713196/8416340639315531)
(8486559416649075/53808689089, 2193799106131211898999306/12481840181908063)
(18278859892840159/79165137769, 4414428225347050503396260/22274140658099147)
(27848579257609359/100465543369, 6996022261432649645861880/31843860022868347)
(50751619624543443/144180443521, 14544822423222140220423054/54746900389802431)
(1466608434929539215/1293194272969, 1792459352454563265743454264/1470603715694798203)
(2933501156515495759/2051090858569, 5047448965487371737114527600/2937496437280754747)
(122154831763316657295/24619727177929, 1350249149361189553521683447304/122158827044081916283)

Commentary

Obtained by a rational 3-isogeny from curve #4751, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/4751. The source commentary attributes the original Mordell curve y^2 = x^3 - 998820191314747 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 = -998820191314747, giving y^2 = x^3 + 26968145165498169. The coordinate map is X = (x^3+4*B)/x^2, Y = y*(x^3-8*B)/x^3. Eleven 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 35915103884809000742847108264324. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 22:59:15 UTC · history