What kind of methods there are to solve a Diophantine equation from IMO longlist?

Question

Namely, in IMO longlist 1987 were given the equation $3z^2=2x^3+385x^2+256x-58195$ and asked to find its integer points. How can I find those? I tried to substitute $z=12k,x=6t$ to get $432k^2=432t^3+13860t^2+1536t-58195.$ But now I'm a bit lost. What to do next? I tried it with Sage:

sage: E=EllipticCurve([0,13860/432,0,1536/432,-58195/432]);E
Elliptic Curve defined by y^2 = x^3 + 385/12*x^2 + 32/9*x - 58195/432 over Rational Field
sage: E.integral_points()
---------------------------------------------------------------------------
ValueError                                Traceback (most recent call last)
<ipython-input-8-d9ed9dfcaaf6> in <module>()
----> 1 E.integral_points()

/usr/lib/sagemath/local/lib/python2.7/site-packages/sage/schemes/elliptic_curves/ell_rational_field.pyc in integral_points(self, mw_base, both_signs, verbose)
   5524         # INPUT CHECK #######################################################
   5525         if not self.is_integral():
-> 5526             raise ValueError, "integral_points() can only be called on an integral model"
   5527 
   5528         if mw_base=='auto':

ValueError: integral_points() can only be called on an integral model

Answer 1

Probably you can this with sage too (check conversion to Weierstrass model).

Using Maple with $y$ instead of $z$:

 with(algcurves);
 wa:=Weierstrassform(2*x^3+385*x^2+256*x-58195-3*y^2,x,y,u,v);

Gave: $ E : {u}^{3}-440067\,u+{v}^{2}-106074110 = 0$ (note you must substitute $u = -u'$ to get a Weierstrass model.

The map from $E$ to your curve is:

$$ x = -1/6 u-385/6$$ $$ y = 1/18 v $$

and $$ u = -6 x-385 $$ $$ v = 18 y $$

Sage found 23 integral points on $E$ and it is isomorphic to your non-integral EC.

Didn't check which integral points map to solutions (if any).

I suppose there is better way to solve this, maybe some property of the RHS.