Projects: here you will find a list of projects
some of which are mandatory for you to try.
Counting curves: here are some histograms showing the
distribution of the number of elliptic curves over Z/(p Z) for p
prime between 5 and 293. This note (ps, pdf) explains why the
histograms are symmetric.
Counting points for different positive characteristics
and experimental evidence for the Birch and SwinnertonDyer
conjecture: here (ps, pdf) is
a short note describing very simple experiments in this direction.
Torsion groups: here (ps, pdf) you can find all groups
that appear as torsion groups of elliptic curves, including a curves
realizing them and some partial frequency data.
Competition! Idea: find an integer point (x,y) on an
elliptic curve y**2 = x**3 + a*x**2 + b * x +c with a, b, c integers
so that R=sqrt(x/discriminant(a,b,c)) is as large as
possible. For example, the point (1,1) is on y**2=x**32*x and we
have R=sqrt(1/discriminant(0,2,0)) = sqrt(1/32) ~ 0.18.
Some better points are:
(x,y)  a  b  c  R  by 
(28186307315582916794821057511400951 0272983487003358,*) 
0  132  935552736240624 
3453575350.92  N. Elkies 
(23330479,112690010143)  7  9 
14  55.5  Anon. 
(17454560,72922784957)  9  4 
9  30.1  Anon. 
(47884,10477737)  4  1 
5  13.6  M. Woodbury 
(5853886516781223,4478849284284020423 07918) 
0  0  1641843  8.97  N. Elkies 
(4677933,10117668338)  12  13 
4  8.08  Anon. 
(3307172,6014313923)  14  4 
7  6.20  Anon. 
(1057027,1086752792)  8  9  8 
5.65  Anon. 
Do you know of other big points? Let us know!.
Cryptography. This is an
interesting overview on applications of elliptic curves to
cryptography.
