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corrected typo in equation
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Timothy Chow
Timothy Chow
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(Collecting comments into a community wiki answer.)

There is a standard method for transforming a smooth cubic into Weierstrass form. See for example Section 1.3 or Appendix B of Silverman and Tate's book Rational Points on Elliptic Curves. It is also implemented in Sage as $\mathtt{EllipticCurve\_from\_cubic}$.

For $n=6$ your curve is isomorphic to $Y^2 - Y = X^3 -54X - 88$$Y^2 + Y = X^3 -54X - 88$. According to the LMFDB, it has torsion $\mathbb{Z}/3\mathbb{Z}$ and rank 1. Thus one can start with the obvious solution $(x:y:z) = (1:2:3)$ and its permutations and generate all rational solutions. Since your equation is homogeneous, finding all rational solutions is equivalent to finding all integer solutions.

The positive solutions constitute one of two connected components of the real locus of the elliptic curve. Since the generator is on that component (and the identity is not), its odd multiples again yield positive solutions. The next batches of positive solutions are permutations of $(1817,3258,5275)$ and $(4904676969, 10840875082, 15051171563)$.

(Collecting comments into a community wiki answer.)

There is a standard method for transforming a smooth cubic into Weierstrass form. See for example Section 1.3 or Appendix B of Silverman and Tate's book Rational Points on Elliptic Curves. It is also implemented in Sage as $\mathtt{EllipticCurve\_from\_cubic}$.

For $n=6$ your curve is isomorphic to $Y^2 - Y = X^3 -54X - 88$. According to the LMFDB, it has torsion $\mathbb{Z}/3\mathbb{Z}$ and rank 1. Thus one can start with the obvious solution $(x:y:z) = (1:2:3)$ and its permutations and generate all rational solutions. Since your equation is homogeneous, finding all rational solutions is equivalent to finding all integer solutions.

The positive solutions constitute one of two connected components of the real locus of the elliptic curve. Since the generator is on that component (and the identity is not), its odd multiples again yield positive solutions. The next batches of positive solutions are permutations of $(1817,3258,5275)$ and $(4904676969, 10840875082, 15051171563)$.

(Collecting comments into a community wiki answer.)

There is a standard method for transforming a smooth cubic into Weierstrass form. See for example Section 1.3 or Appendix B of Silverman and Tate's book Rational Points on Elliptic Curves. It is also implemented in Sage as $\mathtt{EllipticCurve\_from\_cubic}$.

For $n=6$ your curve is isomorphic to $Y^2 + Y = X^3 -54X - 88$. According to the LMFDB, it has torsion $\mathbb{Z}/3\mathbb{Z}$ and rank 1. Thus one can start with the obvious solution $(x:y:z) = (1:2:3)$ and its permutations and generate all rational solutions. Since your equation is homogeneous, finding all rational solutions is equivalent to finding all integer solutions.

The positive solutions constitute one of two connected components of the real locus of the elliptic curve. Since the generator is on that component (and the identity is not), its odd multiples again yield positive solutions. The next batches of positive solutions are permutations of $(1817,3258,5275)$ and $(4904676969, 10840875082, 15051171563)$.

Source Link
Timothy Chow
Timothy Chow
  • 82.6k
  • 26
  • 363
  • 587

(Collecting comments into a community wiki answer.)

There is a standard method for transforming a smooth cubic into Weierstrass form. See for example Section 1.3 or Appendix B of Silverman and Tate's book Rational Points on Elliptic Curves. It is also implemented in Sage as $\mathtt{EllipticCurve\_from\_cubic}$.

For $n=6$ your curve is isomorphic to $Y^2 - Y = X^3 -54X - 88$. According to the LMFDB, it has torsion $\mathbb{Z}/3\mathbb{Z}$ and rank 1. Thus one can start with the obvious solution $(x:y:z) = (1:2:3)$ and its permutations and generate all rational solutions. Since your equation is homogeneous, finding all rational solutions is equivalent to finding all integer solutions.

The positive solutions constitute one of two connected components of the real locus of the elliptic curve. Since the generator is on that component (and the identity is not), its odd multiples again yield positive solutions. The next batches of positive solutions are permutations of $(1817,3258,5275)$ and $(4904676969, 10840875082, 15051171563)$.

Post Made Community Wiki by Timothy Chow