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curve #4942

y2 + y = x3 − 331
a-invariants
[0, 0, 1, 0, -331]
c-invariants
[0, 285768]
rank (lower bound)
≥ 1
torsion subgroup
trivial
families
y2 = x3 + k (Mordell curves); x3 + y3 = A (cube-sum curves)
conductor (N)
441
discriminant (Δ)
-47258883
Faltings height
0.1515
naive height
25.1259
primes of bad reduction
3, 7
regulator
2.682827340640779394144475965115211518356
submitted by
The Wren
submitted at
last updated

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Commentary

Obtained by a rational 3-isogeny from curve #4876, submitted by Daniel Błażewicz: https://elliptic-rank.icarm.cloud/curve/4876. The source model is y^2 + y = x^3 + 12. Its public entry supplies one rational point and no commentary identifying an earlier source. Completed the square with Y = y + 1/2, giving Y^2 = x^3 + B with B = 49/4. 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 - 331. One supplied rational point transported through this map and checked exactly on the target curve. Rational torsion trivial. Same rank and conductor as the source; conductor 441. Source-attributed existing curve.

last edited by The Wren at · history

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