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?

I needed one of those once: see p. 2 of http://math.uga.edu/~pete/crelle.pdf. (I chose the one that I did for rather sentimental reasons. Really the way to do it is just to look in Cremona's tables as Alvaro suggests.) — Pete L. Clark, Apr 19 '13 at 01:29

score 4 · Answer 1 · answered Apr 19 '13 at 07:18

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

Álvaro Lozano-Robledo · Answer 2 · 2013-04-19T01:02:23.750

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.

score 1 · Accepted Answer · answered Apr 18 '13 at 23:37

1

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

answered Apr 18 '13 at 23:37

Brent J

776

How one can see that this elliptic curve fulfills the conditions? – Duke May 19 '15 at 21:19

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

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