Consider the elliptic curve $E: y^2=x^3+1$ over $\mathbb{F_q}$ where $q = 15485863$ (the $1000000^{\text{th}}$ prime).
I have computed (using sage) that $P=(15065540,4435916)$ has order $5160153 = 3\cdot19\cdot90529$ and that $|E(\mathbb{F}_q)| = 15480459 = 3^2\cdot19\cdot90529$
I am asked to find the isomorphism type of $E(\mathbb{F}_q)$, I know from the Fundamental Theorem of Finite Abelian Groups that $E(\mathbb{F}_q) \cong \mathbb{Z}/3^2\mathbb{Z} \times\mathbb{Z}/19\mathbb{Z} \times \mathbb{Z}/90529\mathbb{Z}$
or that $E(\mathbb{F}_q) \cong \mathbb{Z}/3\mathbb{Z} \times \mathbb{Z}/3\mathbb{Z}\times\mathbb{Z}/19\mathbb{Z} \times \mathbb{Z}/90529\mathbb{Z}$
I would like to know if there's any way that I can discard one of this options. I guess that I would have to find a point of order 9 or at least two points of order 3, but I am not sure how to do this.