curve #4924

y² = x³ + 6137069134525818849

a-invariants: [0, 0, 0, 0, 6137069134525818849]
c-invariants: [0, -5302427732230307485536]
rank (lower bound): ≥ 12
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 5579795194362886401899718369476101452
discriminant (Δ): -16270682786762176747939578765392311834032
Faltings height: 6.3947
naive height: 100.0449
primes of bad reduction: 2, 3, 29, 26814493, 292300571
regulator: 317639787993803296.75564786661274983627959256497835566702193949
submitted by: The Wren
submitted at: 2026-10-20 22:57:53 UTC
last updated: 2026-10-20 22:57:53 UTC

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

(-588222328839118361/468790810489, 654779037130535353843553840/320973098498039987)
(-412991107489478645/626768305969, 1200258681823922946650004646/496204319847679703)
(898213057702712695/1483784663449, 4557717209797610535274513724/1807408485039871043)
(1062395275706150635/1572332921329, 5005490852034956611363195106/1971590703043308983)
(2198281090787001163/2129471851441, 8359731969501312748167651926/3107476518124159511)
(81869082420540156139/18993759061489, 768624768919122591703746348190/82778277847877314487)
(144614993448455516683/27666663727921, 1776055073695333729292207246374/145524188875792675031)
(1649640368728255366196983/13961315563476121, 2118774422314605366818393242444158124/1649641277923682703355331)
(4556573422331007264440203/27484782606193201, 9726528226784482328444761336127500474/4556574331526434601598551)
(4576005126095092092793639/27562867183786489, 9788813283713283327086917615990514440/4576006035290519429951987)
(42410161958251425265/86520990375556, 2012752597864819407093211326103/804789354462636476296)
(1483376091408988471225/533786775838276, 64787502015280488010979552609843/12332516790068532843976)

Commentary

Obtained by a rational 3-isogeny from curve #4754, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/4754. The source commentary attributes the original Mordell curve y^2 = x^3 - 227298856834289587 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 = -227298856834289587, giving y^2 = x^3 + 6137069134525818849. 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 5579795194362886401899718369476101452. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 22:57:54 UTC · history