curve #4927

y² = x³ − 33894733080041271

a-invariants: [0, 0, 0, 0, -33894733080041271]
c-invariants: [0, 29285049381155658144]
rank (lower bound): ≥ 11
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 170200434158110225993761639006732
discriminant (Δ): -496304466005049418997808939343630512
Faltings height: 5.5283
naive height: 89.6472
primes of bad reduction: 2, 3, 4136633, 303473981
regulator: 2530554079514346.26701555168502545920711927577961858150306550
submitted by: The Wren
submitted at: 2026-10-20 23:00:27 UTC
last updated: 2026-10-20 23:00:27 UTC

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

(3802125939072355/11413289889, 67614936958410005163626/1219315998711537)
(3877357622095423/10938858921, 118008553081450781346256/1144084315688469)
(4085742311337763/9566600481, 196286775104683875389606/935699626446129)
(4368983119389055/7522613289, 262614043446749057940776/652458818394837)
(5019320608987843/165096801, 355604414550298166965294/2121328796049)
(5051264662038223/961682121, -358962755117076370277756/29822724254331)
(6120642560730319/10650859209, -433981676287213255223740/1099200622946427)
(8558748909398755/23215702689, -450375195007985739483574/3537306971614863)
(17545299957664975/53929379529, 291228941528388480797884/12523858019881083)
(6837921826531224895/3600836422569, 17836462980418759835974931456/6832900384593441003)
(3544974179010743460903235/23249237663678049, 6674513444970224975378759207303871886/3544974173989301523119343)

Commentary

Obtained by a rational 3-isogeny from curve #4752, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/4752. The source commentary attributes the original Mordell curve y^2 = x^3 + 1255360484445973 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 = 1255360484445973, giving y^2 = x^3 - 33894733080041271. 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 170200434158110225993761639006732. Source-attributed existing curve.

last edited by The Wren at 2026-10-20 23:00:27 UTC · history