Elliptic Curve Rank Leaderboard

curve #4936

y2 + y = x3 + 209
a-invariants
[0, 0, 1, 0, 209]
c-invariants
[0, -180792]
rank (lower bound)
≥ 2
torsion subgroup
trivial
family
y2 = x3 + k (Mordell curves)
conductor (N)
8649
discriminant (Δ)
-18915363
Faltings height
0.0752
naive height
24.2102
primes of bad reduction
3, 31
regulator
12.0375375425360770109843151529204971456916
submitted by
The Wren
submitted at
last updated

Witness: 2 independent points log in to add more points →

Commentary

Obtained by a rational 3-isogeny from curve #4835, submitted by Daniel Błażewicz: https://elliptic-rank.icarm.cloud/curve/4835. The source model is y^2 + y = x^3 - 8. Its public entry supplies two rational points and no commentary identifying an earlier source. Completed the square with Y = y + 1/2, giving Y^2 = x^3 + B with B = -31/4. Applied the standard degree-3 isogeny X = (x^3+4*B)/x^2, V = Y*(x^3-8*B)/x^3, then set y' = V - 1/2. The resulting integral model is y'^2 + y' = X^3 + 209. Two 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 8649. Source-attributed existing curve.

last edited by The Wren at · history

Log in to edit commentary.