A formula for the rank over $\mathbb{Q}(t)$ of the elliptic curve $y^2=x^3+At^6+Bt^3+C$

In this paper, we give an explicit formula for the rank and generators (up to finite index) over $\mathbb{Q}(t)$ of all non-trivial elliptic curves of the form $y^2=x^3+At^6+Bt^3+C$, which is a larger class of elliptic surfaces than the one in Kloosterman's paper (2026) and Desjardins and Naskrecki's paper (2024), namely $y^2=x^3+At^6+C$. Our proof provides a new method to find this formula using generators of the geometric Mordell-Weil group even in the case where this geometric Mordell-Weil group (which are $\mathbb{Z}[ω]$-modules) no longer decomposes into rank-one submodules by the methods in Kloosterman's paper (2026) and Desjardins and Naskrecki's paper (2024). Moreover, our proof uses only a few elements in the Galois group of the coordinates of the generators and only light computations by hand.