# How to understand the statement "two (very) general elliptic curves are not isogenous"

Given two elliptic curves $$E_1$$ and $$E_2$$ over $$\mathbb{C}$$.

How to understand the statement "two (very) general elliptic curves are not isogenous"?

Question 1: Should it be "general" or "very general" here?

One way to think about this is to look at pairs $$(E_1,E_2)\in \mathbb{A}_{j_1}\times \mathbb{A}_{j_2}$$ and the set of pairs with $$E_1$$ isogenous to $$E_2$$ consits of finitely (countably) many hypersurfaces in the product. (Here $$\mathbb{A}_j$$ is the $$j$$-line as the coarse moduli of elliptic curves.)

• what do you mean by "very general"?
– ali
Commented Apr 7, 2022 at 11:45
• Maybe I'm totally off, but it seems to me that there should be countably many elliptic curves in a given isogeny class, however there are uncountably many isomorphism classes, and the statement should follow from that. Commented Apr 7, 2022 at 20:50
• This may be helpful: in geometry, an arrangment is in general position if it is not part of a measure zero subset of the space of arrangements (e.g., no three planes intersect at a point). This may be the same notion, except applied to a relevant space. Commented Apr 8, 2022 at 0:34

Question 1: Very general is safest and necessary outside of some specific situations as far as I know. There are some scenarios where it actually can be just general (over a number field, there are only finitely many elliptic curves isogeneous to a given one, but that's kind of hard, see here). I don't think this is the situation over $$\Bbb C$$, but I don't have an immediate reference for you off hand.