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Let $E$ be an elliptic curve over $\mathbb{Q}$. (or over a number field $K$.) If $E$ has two $p$-isogenies over $\mathbb{Q}$, then $E$ has $p^2$ cyclic isogeny over $\mathbb{Q}$.

I want to show it for $p=5$.

This is mentioned in the proof of theorem 2. of Kenku's paper "On the Number of $\mathbb{Q}$-lsomorphism Classes of Elliptic Curves in Each $\mathbb{Q}$-lsogeny Class".

In this case, since the possible images of the mod $5$-Galois representation $G_\mathbb{Q} \to \operatorname{GL}_2 \mathbb{F}_5$ of elliptic curves over the rationals are completely known, using Magma we can compute the possible image of $G_\mathbb{Q} \to \operatorname{GL}_2 \mathbb{Z}/25$ of our $E[25]$, and hence we can compute whether our $E[25]$ has $G_\mathbb{Q}$-invariant cyclic subgroup of order $25$ or not.

But the paper is published in 1980's, when of course we do not have Magma and we do not know the complete classification of the possible image of the Galois representations.

So I think that we can show the highlighted statement more easily.

And related to this, I want to show the following:

Let $E/\mathbb{Q}$ be an elliptic curve with CM. Then $E$ has no $5$-isogenies over $\mathbb{Q}$.

Improve this question
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    $\begingroup$ Q1 is wrong as stated except for $p=2$ and 27a. The first isogeny class of conductor $11$ is a counter -example. The correct (and easy) statement is: If $E\to E'$ and $E\to E''$ are two distinct isogenies of degree $p$ both defined over $\mathbb{Q}$, then there is a cyclic isogeny $E'\to E''$ of degree $p^2$. $\endgroup$
    – Chris Wuthrich
    Commented Sep 11, 2021 at 8:33
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    $\begingroup$ Q2. If an elliptic curve defined over $\mathbb{Q}$ with complex multiplication admits an isogeny of degree $5$ over $\mathbb{Q}$ then it must have bad (additive) reduction at $5$ and the quadratic imaginary field of endomorphisms has discriminant divisible by $5$.. But a quick look at the list reveals that there is no such field of class number $1$. $\endgroup$
    – Chris Wuthrich
    Commented Sep 11, 2021 at 8:38
  • $\begingroup$ @ChrisWuthrich Thanks for your comment. I took a look at a text of elliptic curves, but I couldn't find two statements you mentioned: if an elliptic curve $E/\mathbb{Q}$ has a $5$ -isogeny over $\mathbb{Q}$, then $E$ is bad at $5$. And for a $K$-CM elliptic curve $E$ over $\mathbb{Q}$ and a prime $p$, $p$ is unramified in $K$ iff $E$ is bad at $p$. I found that if $p$ is good, then $p$ is unramified in $K$, but couldn't find the converse. $\endgroup$
    – zom
    Commented Sep 12, 2021 at 8:16
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    $\begingroup$ "If an elliptic curve $E/\mathbb{Q}$ has a 5 -isogeny over $\mathbb{Q}$, then $E$ is bad at 5." is incorrect and it is easy to find counter-examples. I said that if $E$ has CM then this would hold. This follows from the explicit description of the Galois action for good primes. For a cm curve over $\mathbb{Q}$ the image of Galois for a prime of good reduction is the normaliser of either a split or a non-split Cartan subgroup of $\operatorname{GL}_2(\mathbb{F}_p)$. This means there is no $p$-isogeny defined over $\mathbb{Q}$. $\endgroup$
    – Chris Wuthrich
    Commented Sep 12, 2021 at 10:08
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    $\begingroup$ If $p$ is unramified in $K/\mathbb{Q}$, then the reduction type of $E$ over $\mathbb{Q}$ is the same as over $K$. But over $K$ the curve admits good reduction everywhere. All of these statements should not be difficult to find in texts on CM theory. Probably Serre-Tate is a good place to start. $\endgroup$
    – Chris Wuthrich
    Commented Sep 12, 2021 at 10:11

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