(online) Tool to calculate $E(\mathbb{Q})/2E(\mathbb{Q})$ for $E: y^2 = x(x^2 + 3x + 5)$

Question

For some exercise I need to compute the generators of $E(\mathbb{Q})/2E(\mathbb{Q})$, where $E: y^2 = x(x^2 + 3x + 5)$

I did this by the approach from Cassels book 'Lectures on Elliptic Curves' and got the answer $< (0,0), (1,3) >$ which seems reasonable. A friend of mine did the same computation, but did not get $(1,3)$. Therefore we are looking for some (hopefully online) tool that computes this group. However, I know next to nothing of computer algebra systems, so we could use some help.

Answer 1

You could try with Sage, which is open-source and also available online. It has a vast library of functions related to elliptic curves, although I don't know if it can compute $E(\Bbb{Q})/2E(\Bbb{Q})$ directly.

It shouldn't be too difficult to pick up, especially if you know a bit of Python. You can start looking at the tutorial and at the section of the manual devoted to elliptic curves. In particular, be sure to check out how to construct an elliptic curve and the many functions specific for rational curves.

Without further ado, here's a little primer, which should be enough for your purposes (the lines starting with sage: are the input, minus the sage: part).

We can construct your elliptic curve with

sage: x,y = var('x,y')
sage: E = EllipticCurve(y^2 == x*(x^2 + 3*x + 5))

Then we can inquire about its torsion subgroup

sage: E.torsion_subgroup()
Torsion Subgroup isomorphic to Z/2 associated to the Elliptic Curve defined by y^2 = x^3 + 3*x^2 + 5*x over Rational Field

sage: E.torsion_points()
[(0 : 0 : 1), (0 : 1 : 0)]

and about the rank and generators of the free part

sage: E.rank()
1

sage: E.gens()
[(1 : 3 : 1)]

I believe that this should be enough to conclude that indeed $$ E(\Bbb{Q})/2E(\Bbb{Q}) = \langle (0,0), (1,3) \rangle $$