we study the case of curves whose Galois image modulo $2$ is cyclic of order $3$, giving curves of prime conductor but prime square discriminant, as explained above.

Notation and terminology. {#notation-and-terminology. .unnumbered}
-------------------------

$K$ will denote an imaginary quadratic field $\Q(\sqrt{-d})$ of class number $1$, with ring of integers ${{\mathcal{O}}}_K$. As is well known, there are $9$ such fields, with $d\in\{1,2,3,7,11,19,43,67,163\}$. Many of the results of Section \[section 2 torsion\] also apply to the case $K=\Q$.

We say that an ideal or element of ${{\mathcal{O}}}_K$ is *odd* if it is coprime to $2$. Let $e_2$ be the ramification degree of $2$ in $K/\Q$, so that $e_2=1$ except for $K=\Q(\sqrt{-1})$ and $K=\Q(\sqrt{-2})$. Note also that $2$ splits only in $K=\Q(\sqrt{-7})$. The primes dividing $2$ play a crucial role in Section \[section 2 torsion\], where we will denote by $\q$ a prime dividing $2$, and by $\p$ any prime ideal (which might divide $2$ or not). The valuation at $\p$ is denoted $\vp()$. We denote by $\varepsilon$ a generator of the finite unit group ${{\mathcal{O}}}_K^*$ so that $\varepsilon=-1$ except for $K=\Q(\sqrt{-1})$ and $K=\Q(\sqrt{-3})$ when $\varepsilon=\sqrt{-1}$, or $\varepsilon$ is a $6$th root of unity, respectively.

All curves explicitly mentioned will be labeled 