class number of pure cubic fields and elliptic curves

Question

I want to find generators to Mordell Weil group of the Elliptic Curve $y^2=x^3−6321363052$ and class number of $\mathbb Q(\sqrt[3]{6321363052})$. Some suggestions such as algorithm or softwares will be helpful.

Answer 1

You could try SAGE:

sage: K.<x> = NumberField(x^3 - 6321363052)
sage: K.class_number()

sage: E = EllipticCurve(QQ,[0,-6321363052])
sage: E.gens()

EDIT: One should expect the computations to take a while. These are not easy computations to carry out and patience is key.