# Solving 3rd degree Diophantine equation in Sage

Is there a way to find all integer solutions to $y^2=x^3-x/25+9/125$ using Sage? I tried elliptic curves but its command integral_points() won't work as the curve seems not to be an integral model.

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Hello,

It seems that there is no integral solution but I did not find a Sage command to directly answers that.

First of all, EllipticCurve complains because the equations is not defined with integral coefficients (there are rational though). But you can still enumerate the points defined over the rationals. The curve has a group structure, and in that case you are lucky since it is rank 1, i.e. isomorphic to Z (deduced from the E.rank() and E.torsion_order() commands below).

sage: E = EllipticCurve([0,0,0,-1/25,9/125])
sage: E.rank()
1
sage: E.torsion_order()
1


You can then look at the solutions

sage: p = E.gens()
sage: p
(4/25 : 33/125 : 1)
sage: 2*p
(-34319/108900 : -8297279/35937000 : 1)
sage: 3*p
(82995246844/66933451225 : -24050077094058147/17316687833675875 : 1)


And you can check that these triple are solutions of the equation:

sage: x,y,z=p
sage: y**2 == x**3 - 1/25*x + 9/125
True
sage: x,y,z = 2*p
sage: y**2 == x**3 - 1/25*x + 9/125
True


And the theory of elliptic curves tells you that these are all the rational solutions.

In order to prove that there is no integral solution, you need to prove that the denominator gets larger and larger... but I did not know how it can be automatized.

Vincent

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The elliptic curve $E_1: y^2=x^3-x/25+9/125$ is isomorphic to the one $E_2: Y^2=X^3-25X+9\cdot 5^3,$ here we have $X=25x.$ Integral points on $E_1$ are integral points on $E_2.$ The latter can be computed via Sage.

E=EllipticCurve([-25,9*5^3])
E.integral_points()
[(4 : 33 : 1)]


Hence the only candidates are $(4/25 : \pm 33/125 : 1)$ on your curve. Therefore there are no integral points.

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