Is the curve $y^2=x^4+1$ elliptic?
问题内容
The curve $y^2=P(x)$ over the field of complex numbers, where $P(x)$ is a polynomial of degree $4$ without repeating roots, can be transformed with a birational transformation into $Y^2=Q(X)$ with $Q(x)$ of degree $3$ without repeating roots. That is, an elliptic curve. However, if $P(x)$ does not have a rational root, is it still true that the curve $y^2=P(x)$ is equivalent to an elliptic curve over $\mathbb{Q}$? For example, the curve $y^2=x^4+1$. And what does its L-function look like?
回答 (1)
Using Proposition 1.2.1 in the elliptic curve handbook, set $x'=\frac{2(y+1)}{x^2}, y'=\frac{4(y+1)}{x^3}$ , we get the elliptic curve $$(y')^2=(x')^3-4x',$$ whose properties can be found in the LMFDB.