In the paper "Links between Stable elliptic curves and Diophantine equations" for an elliptic curve $E$ with normal Weierstrass form $$y^2 = x^3 -g_2x -g_3$$ with $g_i \in \mathbb{Z}$ wlog. Then he states the Hasse invariant $\delta$ for $E$ is defined as $$\delta_E := -\frac{1}{2} g_2 g_3^{-1} \: (mod \: \mathbb{Q}^{*2}) .$$

In all other references to the Hasse Invariant of elliptic curves, Arithmetic of Elliptic Curves by Silverman etc., it is stated that for an elliptic curve over a finite field the invariant is either 0 or 1.

Furthermore later in the paper Frey begins to talk about the extension of $\mathbb{Q}$ by the square root of the Hasse invariant i.e. $ \mathbb{Q}(\sqrt{\delta_E})$ where $E$ is the 'Frey curve.' In one case this is trivial but in other not.

Are these two separate quantities that I am getting confused, or is there additional structure behind the Hasse invariant which I have not encountered?

In the paper "Links between Stable elliptic curves and Diophantine equations" for an elliptic curve $E$ with normal Weierstrass form $$y^2 = x^3 -g_2x -g_3$$ with $g_i \in \mathbb{Z}$ w.l.o.g. Then he states the Hasse invariant $\delta$ for $E$ is defined as $$\delta_E := -\frac{1}{2} g_2 g_3^{-1} \: (mod \: \mathbb{Q}^{*2}) .$$

In all other references to the Hasse Invariant of elliptic curves, Arithmetic of Elliptic Curves by Silverman etc., it is stated that for an elliptic curve over a finite field the invariant is either $0$ or $1$.

Furthermore later in the paper Frey begins to talk about the extension of $\mathbb{Q}$ by the square root of the Hasse invariant i.e. $ \mathbb{Q}(\sqrt{\delta_E})$ where $E$ is the 'Frey curve.' In one case this is trivial but in other not.

Are these two seperate quantities that I am getting confused, or is there additional structure behind the Hasse invariant which I have not encountered?