  We consider the question: which elliptic curves appear as the Jacobian of a
smooth curve of genus one splitting a Severi--Brauer variety? We provide three
new examples.
  First, we show that if $E$ is any elliptic curve over an algebraically closed
field $k$ and if $F/k$ is a perfect field extension, then there exists a
principal homogeneous space for $E_F$ splitting a Severi--Brauer variety
$X=\mathrm{SB}(A)$ over $F$ if and only if $A$ is Brauer equivalent to a cyclic
algebra. Along the way, we also give a uniform proof of a generalization of
results due to O'Neil, Clark and Sharif, and Antieau and Auel.
  Second, we give an example of an elliptic curve $E$ over a field $k$ together
with a central simple algebra $A/k$ of degree $4$ such that $E$ is the Jacobian
of a smooth genus one curve $C$ embedded in the Severi--Brauer variety
$X=\mathrm{SB}(A)$ as a degree 8 curve and such that $E$ is not the Jacobian of
any genus one curve of smaller degree contained in $X$. Our example is, in some
sense, as small as possible in both dimension and arithmetic complexity.
  Third, we show that for any odd integer $n\geq 3$ there is a central simple
algebra $A$ of degree $n$ over a local field $k$ which is split by the
principal homogeneous space of an elliptic curve $E/k$ but not by any principal
homogeneous space for any quadratic twist of $E$. This generalizes a recent
result of Saltman in the case of surfaces to arbitrary even dimension.
