Specialization of a Mordell curve over Q(T) with generic rank 4.
The curve is a member of a universal parameterized rank-4 family
that includes every rank >= 4 Mordell curve over Q at least once up to isogeny.
Moreover, this family achieves the minimum conductor degree of 12 for a
non-constant Mordell Q(T) curve of generic rank 4,
and accounts for all such degree-12 curves up to Q(T)-isogeny.
The construction is as follows. Choose a rational point (a, b, c, d) on the Burkhardt quartic
1 + a^3 + b^3 + c^3 + d^3 + 3abcd = 0
subject to appropriate non-singularity conditions.
Let
A := a^3 + 1,
R := a^2 d^2 - bc,
H(T) := bT^2 - cT - ad,
G(T) := (b^3 - d^3)T^3 + 3bRT^2 - 3cRT - (2a^3 d^3 + c^3 + d^3).
Define the polynomial
Phi_{a,b,c,d}(T) := A^2/4 (G(T)^2 + 4Ad^3 H(T)^3).
Then the curve
Y^2 = X^3 + Phi_{a,b,c,d}(T)
has generic rank 4 over Q(T), and moreover every rank >= 4 Mordell curve over Q arises as some specialization.
This particular curve arises as (a, b, c, d, T) = (-43/36, -5/12, 1/4, 7/9, 4209911/23085591).
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