Rank $2$ Elliptic Curves

Question

I'm on a quest for some rank $2$ elliptic curves. My question is actually twofold:

• Is there a way to easily construct a curve with this property?

• Is there a database of elliptic curves with given rank?

I know Cremona has an extensive list of curves, but it seems like access to his tables require Linux. Am I wrong on this? References on constructing curves to have a certain rank is welcome, as are Pari/GP and Sage programs that anyone may have.

Answer 1

You can use the Sage Notebook to evaluate Sage code online.

For example:

    elliptic_curves.rank(n=5, rank=2, tors=0, labels=false)

With output:

Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x over Rational Field

Elliptic Curve defined by y^2 + x*y = x^3 + 1 over Rational Field

Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 4*x + 4 over Rational Field

Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 15*x + 16 over Rational Field

Elliptic Curve defined by y^2 + y = x^3 + x^2 - 4*x + 2 over Rational Field

Answer 2

Accessing Cremona's tables does not need Linux:

• You can access the database directly here. For instance, data for the curves of conductors between 1 and 9999 can be found here.

• But nowadays there is a much better interface, the LMFDB database! You can find $100$ elliptic curves of rank $2$ just by searching the database.

Here is the first example with rank $2$ in the database: [0, 1, 1, -2, 0], i.e., $$E: y^2 +y=x^3+x^2-2x.$$