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Let $K$ be a quadratic imaginary field. To simplify my life, let us assume that $K$ has class number one.

Consider the following infinite set:

$S_1:=$ $\{$ $E\subseteq\mathbf{P}^2(\mathbf{C})$ is an embedded elliptic curve defined over $\mathcal{O}_K$ with global minimal Weierstrass equation and with CM by $K$ }

Let $S_2=\{N_{K/\mathbf{Q}}(f_E):E\in S_1 \mbox{where $f_E\subseteq\mathcal{O}_K$ is the conductor of $E/\mathcal{O}_K$ }\}$.

Q1: Is it possible to find an explicit expression $C_K$ (where we think of $K$ as a parameter; forget for the moment that there are only 9 such fields), depending only on $K$ (involving for example the discriminant of $K$), such that there exists always an elliptic curve $E\in S_1$ such that $N_{K/\mathbf{Q}}(f_E)\leq C_K$?

Q2: What do we know about the minimal values in the set $S_2$ ?

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    $\begingroup$ There is a CM elliptic curve (over Q) of conductor $p^2$ for $p=7,11,19,43,67,163$, of conductor $256$ for $d=-8$, and $27$ and $32$ for $d=-3,-4$. I don't know if this helps you or not. I think Greenberg or Rohrlich has a "canonical" Grossencharacter (which is not canonical when $h\neq 1$, maybe need $8\not | d$ also) of infinity type (0,1) and the specified level, namely of norm $\frak p$ for $p=7..163$, and $\frak p^2$ for $d=-3$, and $\frak p^3$ for $d=-4$ and $\frak p^5$ for $d=-8$. $\endgroup$
    – NAME_IN_CAPS
    Commented Jul 15, 2014 at 14:02
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    $\begingroup$ See also the Lemma page 3 of link.springer.com/article/10.1007%2FBF01388657 $\endgroup$
    – NAME_IN_CAPS
    Commented Jul 15, 2014 at 14:08
  • $\begingroup$ Thanks for the data. Yes my questions are closely related to asking for a "canonical Grossencharacter" of a quadratic CM field, but from what you say it seems that such an object does not really exist. $\endgroup$
    – Hugo Chapdelaine
    Commented Jul 15, 2014 at 14:20
  • $\begingroup$ Maybe yes and no for your purposes. I think they all differ by twisting by a class group character, which would preserve the conductor. Rohrlich's paper is projecteuclid.org/euclid.dmj/1077314180 I had the conditions wrong, either $d$ is odd, or $8|d$. In the first case there are $h$ choices, in the second case $2h$. He also takes $d<-4$. He says these canonical characters appear in the book of Gross. $\endgroup$
    – NAME_IN_CAPS
    Commented Jul 15, 2014 at 14:27
  • $\begingroup$ A bit more precisely, an elliptic curve has CM not by $K$ but by some order in $K$, i.e. a finite-index subring of $O_K$. Usually that order must be $O_K$ itself, but there is one more choice for $d=-4$ and $d=-7$, and two more for $d=-3$. In all but one of these four extra cases, specifying the CM ring does not affect the answer, because there's always a minimal-conductor curve with endomorphisms by $O_K$ that's isogenous to $E$ (and isogenies preserve the conductor). But for $d=-3$ there's an index-2 subring ${\bf Z}[\sqrt{-3}]$ of $O_K$ whose minimal conductor is not $27$ but $36$. $\endgroup$
    – Noam D. Elkies
    Commented Jul 15, 2014 at 14:57

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