Find all points of finite order on the elliptic curve $y^2+7xy=x^3+16x$. I am studying Rational Points on Elliptic Curves by Silverman and Tate. This is Problem 2.12 (h).

Determine all of the points of finite order on the elliptic curve $y^2+7xy=x^3+16x$. Also determine the structure of the group formed by these points. 

I have done the other ones that are in Weierstrass standard form with integer coefficients, for example $y^2=x^3-43x+166$. The methods are:


*

*Use the strong Nagell-Lutz theorem. Find $y$ such that $y^2|D$. Try those $y$ values and find the corresponding $x$ values. If they are integers and satisfy the equation of the curve, check if they have finite orders. 

*Use the Reduction Modulo $p$ theorem. Find a prime $p$ such that $p\nmid 2D$. Reduce the equation to modulo $p$. The finite order points on the original curve is a subgroup of the finite order points on the new curve. 


These methods work well for the curves with standard form. But for $y^2+7xy=x^3+16x$, I am not sure what to do. I can make change of variable, of course, to transform it into $Y^2=x^3+\frac{49}{4}x^2+16x$. But this introduce a fraction coefficient, also the change of variable involves a fraction $Y=y+\frac{7}{2}x$. With this the reduction modulo $p$ would not work. Even the N-L theorem is about integer division. 
My question:
How can I get rid of the fractions? Or is there another way to find the finite order points?
Thank you for any help!

Your curve $y^2 + 7xy = x^3 + 16x$ is isomorphic to $y^2 = x^3 - 44091x + 3304854$. Maybe you can find the transformation between these two models yourself?
