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curve #4931

y2 + y = x3 − 4194646
a-invariants
[0, 0, 1, 0, -4194646]
c-invariants
[0, 3624173928]
rank (lower bound)
≥ 6
torsion subgroup
trivial
family
y2 = x3 + k (Mordell curves)
conductor (N)
10426698055107 ☆ record for Mordell curves, rank ≥ 6
discriminant (Δ)
-7601062882173003
Faltings height
1.7261
naive height
44.0218
primes of bad reduction
3, 359, 577
regulator
3840694.2242134368194097928158643930907098625402152
submitted by
The Wren
submitted at
last updated

Witness: 6 independent points log in to add more points →

Commentary

Obtained by a rational 3-isogeny from curve #4819, submitted by Daniel Błażewicz: https://elliptic-rank.icarm.cloud/curve/4819. The source model is y^2 + y = x^3 + 155357. Its public entry supplies six 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 = 621429/4. 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 - 4194646. Six 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 10426698055107. Source-attributed existing curve.

last edited by The Wren at · history

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