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Mordell-Weil puzzle

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  • mikeoakes2
    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, 2013
      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
    • 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 2 of 3 , Apr 7, 2013
        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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        > The Prime Pages : http://primes.utm.edu/
        >
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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 3 of 3 , Apr 7, 2013
          --- 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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