GonzálezJiménez and Xarles studied a problem in Diophantine number theory and they obtained several nice results via elliptic curve Chabauty's method over quadratic number fields. At page 73 in paper there are six 5tuples mentioned where they could not determine the set of rational solutions. Trying to do something over degree 4 composite fields I was able to handle some of the remaining cases, but not the tuple $(0,2,5,7,11).$ One possible genus 2 curve to attack this case is $y^2=5x^6  16x^4  19x^2 + 66,$ so it is bielliptic the rank of the Jacobian is 2 and there are 8 torsion points. The degree six polynomial is reducible: $5(x^23)(x^2+2)(x^211/5).$ That is there are some degree 4 number fields suggested by the factors. I could not make use of those 3 fields. Is there any argument to choose a quadratic number field, let say NumberField($x^2\pm d$) and one of the 3 suggested fields by the factorization and work in the composite quartic fields?

1$\begingroup$ As far as I see there are points: $(1:\pm 6:1)$ and $(1:\pm 6:1).$ $\endgroup$ – castor Oct 8 '19 at 14:55
You can apply the socalled Elliptic Chabauty over the biquadratic field $K:=\mathbb{Q}(\sqrt{3},\sqrt{11/5})$ (also equal to the field adjoining a root of $ 25x^4  260x^2 + 16$). Over this field there are two possible 2coverings (one corresponding to the points with coordinate $x=1$, the other with coordinate $x=1$). Both curves have a map to a genus one curve given by an equation $$ \delta_{\pm} y^2=(x\sqrt{3})(x\sqrt{11/5})(x^22)$$ with $$\delta_{\pm}=(\pm 1\sqrt{3})(\pm 1\sqrt{11/5})$$ (one for every sign). Both cases we get an elliptic curve with rank 2, so we can apply Elliptic Chabauty MAGMA function, which answers that the only points with rational $x$coordinate are the ones with $x=\pm 1$.
You can do (attempt) etale descent over one of the quadratic number fields defined by any of the factors. See section 8.3 of my paper Elliptic curves over $\mathbb Q$ and 2adic images of Galois with Jeremy Rouse for an example of how to do this (there is code available on the arXiv too).