We determine the probability that a randomly chosen elliptic curve $E/{F}_p$ over a randomly chosen prime field ${F}_p$ has an ${ell}$-primary part $E({F}_p) [ell^{infty}]$ isomorphic with a fixed abelian $ell$-group $H^{(ell)}_{lpha,eta} = {Z}/{ell}^{lpha} imes {Z}/ell^{eta}$. smallskip Probabilities for ``$|E(F_p)|$ divisible by $n$'', ``$E(F_p)$ cyclic'' and expectations for the number of elements of precise order $n$ in $E(F_p)$ are derived, both for unbiased $E/F_p$ and for $E/F_p$ with $p equiv 1~(ell^r)$.
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机译：我们确定在随机选择的素数字段$ { F} _p $上随机选择的椭圆曲线$ E / { F} _p $具有$ { ell} $主要部分$ E（{ F} _p）[ ell ^ { infty}] $同构，带有固定的阿贝尔$ ell $ -group $ H ^ {（ ell）} _ { alpha， beta} = { Z} / { ell} ^ { alpha} times { Z} / ell ^ { beta} $。 smallskip``$ | E（ F_p）| $可被$ n $整除''，``$ E（ F_p）$循环''的概率以及对$中精确顺序$ n $的元素数量的期望导出E（ F_p）$，对于无偏的$ E / F_p $和带有$ p equiv 1〜（ ell ^ r）$的$ E / F_p $。
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