curve #4757

y² + y = x³ − 33157242143901824764642

a-invariants: [0, 0, 1, 0, -33157242143901824764642]
c-invariants: [0, 28647857212331176596650472]
rank (lower bound): ≥ 14
torsion subgroup: trivial
family: y² = x³ + k (Mordell curves)
conductor (N): 651497900201089941708666232118910606092447907 ☆ record for Mordell curves, rank ≥ 14
discriminant (Δ): -474941969246594567505617683214685831841394524203
Faltings height: 7.8272
naive height: 117.2342
primes of bad reduction: 3, 293, 7459, 170389, 4397072149
regulator: 466836721365196397143.675184261737252540077379450724508198027766919
submitted by: David Renshaw
submitted at: 2026-10-20 21:28:41 UTC
last updated: 2026-10-20 21:28:41 UTC

Witness: 14 independent points log in to add more points →

(3684219550495548478522/114671950833001, 3155910407433421859339478925078/1227964470823240375499)
(4227163697499665779798/77708579671009, 244899346376000172212105404901091/685020323819123074223)
(280486158834371611834969/4189272550668100, 140102693295948356330094754314930953/271148948240246998579000)
(4603713719706608396272/45653657049001, 307273323778754641469239287195753/308470301612180457749)
(4639997157962970900778/41999563219849, 312154902765447683743866119907631/272186863355817953243)
(4909761412274719447828/1803802105249, 344025175746277870493358613610101/2422609044069406193)
(4912171847175891512896/52920702025, 344278837299459083064594290788569/12174142897341125)
(4912295862846616019422/232132203601, -344291874595171968366272206700763/111841527827165401)
(5068652940300290899822/29037020737201, -359733539668199658773767092368158/156468918981502045801)
(5414786305003136268772/63214441464001, -387795347387745208960456575185253/502602283684347414751)
(26446263442651037158738/774017667095929, 1766658457155680044197162310013754/21534079421332248304717)
(370665442800050788613788/5114369908721329, 215618098289833713375089172424041654/365753258778731999759767)
(35645694694032170001956497/430691722569113476, 206501811769628268719238019451965571293/282650519333342140122393224)
(2436974646091584272932682896/1810923440757858025, 120302197879045539715675370836789830415806/2436969733907562954143828875)

Commentary

Mordell curve y^2 = x^3 + -2122063497209716784937072. Source: N. D. Elkies, 'j=0, rank 15; also 3-rank 6 and 7 in real and imaginary quadratic fields', NMBRTHRY post, 30 Dec 2009, https://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;6a3fad67.0912&S=. Points are images of Elkies' points on y^2 = x^3 + 78594944341100621664336 under the 3-isogeny (x,y) -> ((x^3+4k)/x^2, y(x^3-8k)/x^3).

last edited by David Renshaw at 2026-10-20 21:28:41 UTC · history