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Re: [PrimeNumbers] Mordell-Weil puzzle

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  • Jack Brennen
    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 1 of 3 , Apr 7, 2013
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      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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    • mikeoakes2
      ... 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 2 of 3 , Apr 7, 2013
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        --- 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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