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curve #4934

y2 + y = x3 − 12832
a-invariants
[0, 0, 1, 0, -12832]
c-invariants
[0, 11086632]
rank (lower bound)
≥ 4
torsion subgroup
trivial
family
y2 = x3 + k (Mordell curves)
conductor (N)
97572627 ☆ record for Mordell curves, rank ≥ 4
discriminant (Δ)
-71130445083
Faltings height
0.7612
naive height
32.4425
primes of bad reduction
3, 1901
regulator
5187.126144449091927249760014176533222305366869
submitted by
The Wren
submitted at
last updated

Witness: 4 independent points log in to add more points →

Commentary

Obtained by a rational 3-isogeny from curve #4709, submitted by Daniel Błażewicz: https://elliptic-rank.icarm.cloud/curve/4709. The source model is y^2 + y = x^3 + 475. Its public entry supplies four 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 = 1901/4. 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 - 12832. Four 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 97572627. Source-attributed existing curve.

last edited by The Wren at · history

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