The Generalized Elliptic Curves $(GECs)$ are pairs $(Q,T)$, where $T$ is a family of triples $(x,y,z)$ of “points” from the set $Q$ characterized by equalities of the form $x.y=z$, where the law $x.y$ makes $Q$ into a totally symmetric quasigroup. Isotopic loops arise by setting $x*y=u.(x.y)$. When $(x.y).(a.b)=(x.a).(y.b)$, identically $(Q,T)$ is an entropic $GEC$ and $(Q,*)$ is an abelian group. Similarly, a terentropic $GEC$ may be characterized by $x^2.(a.b)=(x.a)(x.b)$ and $(Q,*)$ is then a Commutative Moufang Loop $(CML)$. If in addition $x^2=x$, we have Hall $GECs$ and $(Q,*)$ is an exponent $3$

We prove that for any $t\in 𝐐$, the curve$$5x^3+9y^3+10z^3+12{\left(\frac{t^2+82}{t^2+22}\right)}^3{(x+y+z)}^3=0$$in ${𝐏}^2$ is a genus $1$ curve violating the Hasse principle. An explicit Weierstrass model for its jacobian $E_t$ is given. The Shafarevich-Tate group of each $E_t$ contains a subgroup isomorphic to $𝐙/3\times 𝐙/3$.