  In this paper we study the arithmetic and invariant theory of genus one
normal curves embedded in $\mathbb{P}^{n-1}$. We generalize the notion of genus
one model of degree $n$, introduced by Cremona, Fisher and Stoll for $n \leq
5$, to arbitrary odd $n$, and describe the invariant theory of a genus one
curve of degree $n$ embedded in $ \mathbb{P}^{n-1}$ in terms of the minimal
graded free resolution of its homogeneous ideal. We prove that everywhere
locally soluble genus one curves over $ \mathbb{Q}$ admit minimal integral
models, with the same invariants as those of the minimal model of their
Jacobian elliptic curve. We then apply these results to study the capitulation
problem for the Tate-Shafarevich group of an elliptic curve $E/\mathbb{Q}$. We
prove that every element of $\text{Sha}(E/\mathbb{Q})[n]$ of odd index $n$
splits over a degree $n$ number field $K$, of absolute discriminant at most
$c(n) H_E^{2n-2}$, where $H_E$ is the naive height of $E$ and $c(n)$ is a
constant only depending on $n$.
