- 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 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

>

>

>

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>

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> The Prime Pages : http://primes.utm.edu/

>

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> - --- In primenumbers@yahoogroups.com, Jack Brennen <jfb@...> wrote:
>

Of course you are quite right, Jack.

> 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.

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