curve #3642

y² + xy = x³ − 26844785415316950775425605x + 53078430978883527280513999906842557121

a-invariants: [1, 0, 0, -26844785415316950775425605, 53078430978883527280513999906842557121]
rank (lower bound): ≥ 14
torsion subgroup: ℤ/2ℤ
conductor (N): 52730095526316529723237209878616107447931486
discriminant (Δ): 21029407578843431715833967440221498961177210955277982484181090108410515423232
Faltings height: 13.4641
naive height: 187.2699
primes of bad reduction: 2, 3, 7, 17, 23, 31, 37, 43, 53, 59, 73, 211, 601, 36697, 78193, 10736721060361
regulator: 13120789678363475.4055177731366404580001704564904550327069841448697
submitted by: Michael Rubinstein
submitted at: 2026-09-29 19:25:25 UTC
last updated: 2026-09-29 19:25:25 UTC

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

(2744993348530, 270492323384302471)
(3588059963734, 1717857507293552395)
(-6892997802664256690/1164241, 2658434546377179804569101001/1256216039)
(-20170153738118276/6241, 5078073203933881759673545/493039)
(-130923827481359/64, 5105479194265758846155/512)
(10957461718429/16, 378711332743068346099/64)
(43942428668157586/5041, 7853225112404850818324321/357911)
(52105349524128538/19321, 1462089790391458907311597/2685619)
(274976860125142750/127449, 103784034551569266067708723/45499293)
(954201995338712866/290521, 91233583613669663732249909/156590819)
(1080022111679406876/326041, 131932424921627117587421973/186169411)
(113167514087470352530/52171729, 847216983197427929671718419793/376836398567)
(163452520385516912530/707281, 2089195959565796959709499352879/594823321)
(3016743851994613779682/927994369, 11482631822057360799205984302361/28269492462847)

Commentary

Specialization at t = -60/91, u = 2/5 of the Elkies-Klagsbrun family y^2 = x^3 + 2A(t,u)x^2 + B(t,u)x (arXiv:2003.00077, Sec. 9, eqs. (3)-(4)), found by searching small-height t = a/b for small conductor, with points from a 2-descent (PARI ellrank).

last edited by Michael Rubinstein at 2026-09-29 19:25:25 UTC · history