One has the notion of a compact Riemann surface with "many automorphisms", a concept for which there exist many equivalent definitions. You have correctly identified the "genus one Riemann sufaces with many automorphisms" -- a situation which is slightly complicated by the fact that, strictly speaking, the automorphism group of any complex elliptic curve as a Riemann surface is infinite -- so let's concentrate on the case of Riemann surfaces $X$ of genus $g \geq 2$, in which case the automorphism group is always finite.

The following three conditions on $X$ are equivalent:

(1) On the moduli space $\mathcal{M}_g$ of genus $g$ complex algebraic algebraic curves, the function $C \mapsto \# \operatorname{Aut}(C)$ has a strict local maximum at $X$ -- explicitly, there exists a neighborhood of $U$ of $X$ such that every curve in $U \setminus \{X\}$ has smaller automorphism group than $X$.

(2) The natural map $X \rightarrow X/\operatorname{Aut}(X)$ is a Belyi map -- i.e., the quotient has genus zero and the map is ramified only over three points. By the "easy" direction of Belyi's theorem, it follows that $X$ can be defined over some number field, and it is interesting to study the field of moduli of such curves $X$ (which in this case is known to be the minimal field of definition).

(3) $X$ is uniformized by a finite index torsion free subgroup $\Gamma$ of a hyperbolic triangle group $\Delta(a,b,c)$. This makes clear that these curves are a significant generalization of the Hurwitz curves, which correspond to the "first" hyperbolic triangle group $\Delta(2,3,7)$. At times I have expressed the opinion that it is strange that there has been so much research done on the $\Delta(2,3,7)$ case -- i.e., when the automorphism group is numerically as large as it could possibly be -- rather than on the more general case of $\Delta(a,b,c)$, i.e., the case where the automorphism group is sufficiently large to have the above very interesting analytic / topological / arithmetic implications.

For more information on these Riemann surfaces, see the bibliography of

http://math.uga.edu/~pete/triangle-091309.pdf

Especially highly recommended is [50], a paper by J. Wolfart. In general Wolfart is one of the pioneers of this subject and has written many interesting papers, mostly from the perspective of complex function theory.

One has the notion of a compact Riemann surface with "many automorphisms", a concept for which there exist many equivalent definitions. You have correctly identified the "genus one Riemann sufaces with many automorphisms" -- a situation which is slightly complicated by the fact that, strictly speaking, the automorphism group of any complex elliptic curve as a Riemann surface is infinite -- so let's concentrate on the case of Riemann surfaces $X$ of genus $g \geq 2$, in which case the automorphism group is always finite.

The following three conditions on $X$ are equivalent:

(1) On the moduli space $\mathcal{M}_g$ of genus $g$ complex algebraic algebraic curves, the function $C \mapsto \# \operatorname{Aut}(C)$ has a strict local maximum at $X$ -- explicitly, there exists a neighborhood of $U$ of $X$ such that every curve in $U \setminus \{X\}$ has smaller automorphism group than $X$.

(2) The natural map $X \rightarrow X/\operatorname{Aut}(X)$ is a Belyi map -- i.e., the quotient has genus zero and the map is ramified only over three points. By the "easy" direction of Belyi's theorem, it follows that $X$ can be defined over some number field, and it is interesting to study the field of moduli of such curves $X$ (which in this case is known to be the minimal field of definition).

(3) $X$ is uniformized by a finite index torsion free subgroup $\Gamma$ of a hyperbolic triangle group $\Delta(a,b,c)$. This makes clear that these curves are a significant generalization of the Hurwitz curves, which correspond to the "first" hyperbolic triangle group $\Delta(2,3,7)$. At times I have expressed the opinion that it is strange that there has been so much research done on the $\Delta(2,3,7)$ case -- i.e., when the automorphism group is numerically as large as it could possibly be -- rather than on the more general case of $\Delta(a,b,c)$, i.e., the case where the automorphism group is sufficiently large to have the above very interesting analytic / topological / arithmetic implications.

For more information on these Riemann surfaces, see the bibliography of

http://alpha.math.uga.edu/~pete/triangle-091309.pdf

Especially highly recommended is [50], a paper by J. Wolfart. In general Wolfart is one of the pioneers of this subject and has written many interesting papers, mostly from the perspective of complex function theory.