Definition of characters in the context of twists of elliptic curves In the Wikipedia article about Twists of curves, I encountered the term character several times and I do not know what the definition of it is (and there is no reference to another Wikipedia article explaining this definition). To cite the relevant passages of the article:

It is possible to "twist" elliptic curves with j-invariant equal to 1728 by quartic characters; twisting a curve E by a quartic twist, one obtains precisely four curves: one is isomorphic to E, one is its quadratic twist, and only the other two are really new. Also in this case, twisted curves are isomorphic over the field extension given by the twist degree.


Analogously to the quartic twist case, an elliptic curve over {\displaystyle K}K with j-invariant equal to zero can be twisted by cubic characters. The curves obtained are isomorphic to the starting curve over the field extension given by the twist degree.

Could you please explain me the definition of the character in this context?
 A: Let $E$ be an elliptic curve over a (perfect) field $K$ and let $G_K = \mathrm{Gal}(\overline K/K)$. As explained in Chapter X.2 of Silverman's Arithmetic of Elliptic Curves, the twists of $E$ are in bijective correspondence with elements of $H^1(G_K, \mathrm{Aut}(E))$.
In general, $\mathrm{End}(E) = \mathbb Z$, so $\mathrm{Aut}(E) = \{\pm1\}$. Since $G_K$ acts trivially on $\{\pm1\}$, in this case,
$$H^1(G_K, \mathrm{Aut}(E)) = \mathrm{Hom}(G_K, \{\pm1\})$$
i.e. the twists of $E$ are in bijective correspondence with quadratic Galois characters $G_K\to\{\pm 1\}$. Alternatively, by Kummer theory, $H^1(G_K,\{\pm 1\})\cong K^\times/K^{\times 2}$, which explains why we usually talk about twisting by an element $d\in K^\times$. The character $\chi\colon G_K\to\{\pm 1\}$ corresponding to $d$ is exactly the lift of the canonical character $\mathrm{Gal}(K(\sqrt d)/K)\to \{\pm 1\}$.
If $\mathrm{Aut}(E)$ is strictly larger than $\{\pm1\}$, there will be extra types of twists. When $K$ is a number field, since $\mathrm{End}(E)$ is at worst an order in an imaginary quadratic field, the only way in which $E$ can gain extra automorphisms is if $\mathrm{End}(E) = \mathbb Z[i]$ or $\mathbb Z[\zeta_3]$. This happens exactly for the elliptic curves of $j$-invariant $1728$ and $0$.
In the case of a $j$-invariant $1728$ curve (e.g. take $E\colon y^2 = x^3 - x$), we have $\mathrm{End}(E) = \mathbb Z[i]$, so $\mathrm{Aut}(E) = \{\pm 1, \pm i\} = \mu_4$. Kummer theory gives an isomorphism
$$H^1(G_K, \mu_4)\cong K^\times/K^{\times 4}$$
and the twist corresponding to $d\in  K^\times/K^{\times 4}$ is $E_d\colon y^2 = x^3 - dx$. Note that, over $K(\sqrt[4]d)$, the map $(x, y)\mapsto (\sqrt dx,\sqrt[4]{d^3}y)$ identifies $E$ and $E_d$. In particular, if $d$ is a square, then $E_d$ is just a quadratic twist.
If $K\supset \mathbb Q(i)$, then $G_K$ acts trivially on $\mu_4$, and again $H^1(G_K, \mu_4)\cong\mathrm{Hom}(G_K, \mu_4)$ so that we get a bijection between twists and quartic Galois characters $G_K\to\mu_4$. The character corresponding to $d\in K^{\times}$ is exactly the map $\sigma\mapsto \frac{\sigma(\sqrt[4]d)}{\sqrt[4]d}\in\mu_4$.
In the case of $j$-invariant $0$ (e.g. $E\colon y^2 = x^3 + 1$), $\mathrm{End}(E) = \mathbb Z[\zeta_3] = \mathbb Z[\zeta_6]$ and you get sextic twists and twists by sextic characters $G_K\to\mu_6$ (if $K\supset \mathbb Q(\zeta_3)$) in the same way. I don't know why the Wikipedia article focuses on just the cubic twists in this case.
