Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
Mathic - it probably originated with me giving a talk / demo of how sage is capable of calling out to other systems, and somebody got the wrong impression. Anyway the goal of sage is to be a free viable alternative to Magma (Mathematica, etc.), so calling them for core functionality would be counterproductive.
Chapter 5 of Silverman-Tate is about integral points, and about why there are finitely many. It doesn't give much help in actually determining them, which was the question.
Having seen many beginners learn about the descent algorithm, both from the perspective of Silverman-Tate and Galois cohomology, I think it's probably better for even a beginner to just learn Galois cohomology and learn descent the right way. All the notation and equations in the presentation in Silverman-Tate really obscure what is going on, in my experience.
"You haven't heard the one about showing Birch-Swinnerton-Dyer unprovable by showing the rank is not computable?" Huh???! What the heck are you talking about? Assuming rank(E)<=1 or Sha(E)[p^oo] finite for one p, or BSD rank is true, then there is a deterministic algorithm to compute E(Q). Without making one of those assumptions, we still don't know.
Here's an example of finding a point on a rank one curve in Sage using Heegner points: "E = EllipticCurve('37a'); P = E.heegner_point(-7); P.point_exact()". The rank 1 curve $y^2=x^3+7823$ provides a spectacular example in which Heegner points fail in practice, but doing a FOUR descent succeeds. Michael Stoll wrote a paper about this, but I can't find it online anymore (it vanished from his website), so here is a temporary link: wstein.org/home/wstein/tmp/4-descent.pdf Also, having read the ancient Peter Green Heegner points program you link above, I do not recommend it.
A very recent version of ratpoints is in Sage. Inputing "import sage.libs.ratpoints as r; r.ratpoints([46224, -3024, 0, 1], 200)" will output lots of points on the curve $y^2 = x^3 - 3024x + 46224$, such as [(1, 0, 0), (-60, 108, 1), (-60, -108, 1), (-32, 332, 1), ...]
To compute generators of the Mordell-Weil group of $y^2+a_1xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6$, type EllipticCurve([a1,a2,a3,a4,a6]).gens(). If you type print(EllipticCurve([a1,a2,a3,a4,a6]).mwrank()), then you'll see what mwrank did. For more about elliptic curves over the rationals in Sage, see this page: sagemath.org/doc/reference/sage/schemes/elliptic_curves/…
Regarding "Sage over the standalone version of mwrank": In fact, Cremona has even deprecated the standalone version. See the note here: warwick.ac.uk/~masgaj/mwrank Also, it is worth mentioning Denis Simon's algebraic 2-descent code, which is also in Sage, and can be fast in certain case.
