  Determinants with Legendre symbol entries have close relations with character
sums and elliptic curves over finite fields. In recent years, Sun, Krachun and
his cooperators studied this topic. In this paper, we confirm some conjectures
posed by Sun and investigate some related topics.
  For instance, given any integers $c,d$ with $d\ne0$ and $c^2-4d\ne0$, we show
that there are infinitely many odd primes $p$ such that
$$\det\bigg[\left(\frac{i^2+cij+dj^2}{p}\right)\bigg]_{0\le i,j\le p-1}=0,$$
where $(\frac{\cdot}{p})$ is the Legendre symbol. This confirms a conjecture of
Sun.
