## Mordell-Weil puzzle

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• Find a rational point on the curve y^2 = x^3 - 6077 Hint: there is exactly one. Comment: the rationale for posting to this list is that prime numbers play a
Message 1 of 3 , Apr 7 1:18 PM
Find a rational point on the curve
y^2 = x^3 - 6077

Hint: there is exactly one.

Comment: the rationale for posting to this list is that prime numbers play a fundamental role in the algebra of elliptic curves.

-Mike Oakes
• The set of rational points is finitely generated, but there are definitely multiple rational points on this curve. Once you find one solution x,y, you can get
Message 2 of 3 , Apr 7 2:30 PM
The set of rational points is finitely generated, but there are definitely multiple rational points on this curve.

Once you find one solution x,y, you can get more using PARI/GP:

x=SOLUTION_X
y=SOLUTION_Y
e=ellinit([0,0,0,0,-6077])
print(ellpow(e,[x,y],2))
print(ellpow(e,[x,y],3))
print(ellpow(e,[x,y],4))
print(ellpow(e,[x,y],5))

On 4/7/2013 1:18 PM, mikeoakes2 wrote:
> Find a rational point on the curve
> y^2 = x^3 - 6077
>
> Hint: there is exactly one.
>
> Comment: the rationale for posting to this list is that prime numbers play a fundamental role in the algebra of elliptic curves.
>
> -Mike Oakes
>
>
>
> ------------------------------------
>
> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
> The Prime Pages : http://primes.utm.edu/
>
>
>
>
>
>
• ... Of course you are quite right, Jack. Brain not in gear :-( I was meaning: the rank is one, i.e. there is only one generator (and all other points are
Message 3 of 3 , Apr 7 3:11 PM
--- In primenumbers@yahoogroups.com, Jack Brennen <jfb@...> wrote:
>
> The set of rational points is finitely generated, but there are definitely multiple rational points on this curve.
>
> Once you find one solution x,y, you can get more using PARI/GP:
>
> x=SOLUTION_X
> y=SOLUTION_Y
> e=ellinit([0,0,0,0,-6077])
> print(ellpow(e,[x,y],2))
> print(ellpow(e,[x,y],3))
> print(ellpow(e,[x,y],4))
> print(ellpow(e,[x,y],5))
>
> On 4/7/2013 1:18 PM, mikeoakes2 wrote:
> > Find a rational point on the curve
> > y^2 = x^3 - 6077
> >
> > Hint: there is exactly one.

Of course you are quite right, Jack.
Brain not in gear :-(

I was meaning: the rank is one, i.e. there is only one "generator" (and all other points are simply integral multiples of that, as per your code).

Mike
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