On elliptic curves and arithmetical graphs

Brumer and Kramer give sufficient criteria to conclude for a givenprime $p$ the non-existence of an elliptic curve $E/mathbb{Q}$ of conductor $p$.Some of these criteria arise out of how primes factor in the $2$-division and $3$-division fields of the elliptic curve. In this paper we take a similar approach except instead of $mathbb{Q}$ our base field is any one of the (exactly 9) class number 1 quadratic imaginary number fields. For a certain $6$ of these number fieldswe are able, in each case, to exhibit a long list of prime numbers less than 500 that are residual characteristics of prime ideals for which we have a non-existence result.We then relate these non-existence results to a conjecture of Cremona.A new invariant, called the $g$-integer, of an arithmetical graph is introduced by Lorenzini. Here we determine the effect on this invariant under basic operations on the arithmetical graph. We then focus on the case of arithmetical trees whose $g$-integer is 0 or 1. Moreover, we computethis invariant for certain modular curves.