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curve #4937

y2 + y = x3 − 19690
a-invariants
[0, 0, 1, 0, -19690]
c-invariants
[0, 17011944]
rank (lower bound)
≥ 4
torsion subgroup
trivial
family
y2 = x3 + k (Mordell curves)
conductor (N)
229740003
discriminant (Δ)
-167480462187
Faltings height
0.8326
naive height
33.2989
primes of bad reduction
3, 2917
regulator
6263.456972140580485722736507855081408989093685
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 #4794, submitted by Daniel Błażewicz: https://elliptic-rank.icarm.cloud/curve/4794. The source model is y^2 + y = x^3 + 729. 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 = 2917/4. 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 - 19690. 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 229740003. Source-attributed existing curve.

last edited by The Wren at · history

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