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Inspired by methods of N. P. Smart, we describe an algorithm to determine all Picard curves over Q with good reduction away from 3, up to Q-isomorphism. A correspondence between the isomorphism classes of such curves and certain quintic binary forms possessing a rational linear factor is established. An exhaustive list of integral models is determined, and an application to a question of Ihara is discussed.
We report on the construction of a database of nonhyperelliptic genus 3 curves over Q of small discriminant.
Let $C$ be a smooth, absolutely irreducible genus-$3$ curve over a number field $M$. Suppose that the Jacobian of $C$ has complex multiplication by a sextic CM-field $K$. Suppose further that $K$ contains no imaginary quadratic subfield. We give a bo
We study the sequence of zeta functions $Z(C_p,T)$ of a generic Picard curve $C:y^3=f(x)$ defined over $mathbb{Q}$ at primes $p$ of good reduction for $C$. We define a degree 9 polynomial $psi_fin mathbb{Q}[x]$ such that the splitting field of $psi_f
We describe normal forms and minimal models of Picard curves, discussing various arithmetic aspects of these. We determine all so-called special Picard curves over $mathbb{Q}$ with good reduction outside 2 and 3, and use this to determine the smalles
An elliptic curve $E$ over $mathbb{Q}$ is said to be good if $N_{E}^{6}<max!left{ leftvert c_{4}^{3}rightvert ,c_{6}^{2}right} $ where $N_{E}$ is the conductor of $E$ and $c_{4}$ and $c_{6}$ are the invariants associated to a global minimal model of