Example of an elliptic curve with trivial torsion subgroup and rank 0

Question

What is an example of an elliptic curve over $\mathbb{Q}$ with trivial torsion subgroup and rank 0?

Answer 1

Do you just need a single example? In that case $y^2+2y=x^3+2x+1$ works

Answer 2

If you need a source of examples, you should look through the "Elliptic Curve Data" by John Cremona (et al.).

For example, at this link you can find a list of curves of conductor between $0$ and $9999$, and the last two digits are the rank, and the order of the torsion subgroup. So you are looking for curves whose line in that table ends in "] 0 1". Searching with my browser I find $11065$ such curves:

• 11 a 2 [0,-1,1,-7820,-263580] 0 1

• 19 a 2 [0,1,1,-769,-8470] 0 1

• 26 a 2 [1,0,1,-460,-3830] 0 1

• 26 b 2 [1,-1,1,-213,-1257] 0 1

• ...

Etc.

Answer 3

As Alvaro notes, Cremona's tables are a definite source. Following the link on Cremona's web site to lmfdb.org, you can search more easily, and the exact search for curves with trivial torsion and rank 0 can be encoded in the following URL

http://www.lmfdb.org/EllipticCurve/Q?conductor=&rank=0&torsion=1