Corollary 6.4.Leave $ E $ be an elliptical curve and leave $ m en mathbb {Z} $ with $ m ne $ 0.(b) yes $ m ne $ 0 in $ K $, that is, if any $ operatorname {char} (K) = 0 $ or $ p: = operatorname {char} (K)> 0 $ Y $ p nmid m $, so $ E[m] = mathbb {Z} / m mathbb {Z} times mathbb {Z} / m mathbb {Z} $.

(page 86 of [Silverman, The Arithmetic of Elliptic Curves, 2nd Edition])

Leave $ E: y ^ 2 = x ^ 3 + 3x + 8 $ finished $ mathbb {Z} / 13 mathbb {Z} $. I just calculated the points on this curve. exist $ # E[3] stackrel {?} {=} 2 $ order points $ 3 $ Y $ # E[9] stackrel {?} {=} 6 $ Points of order 9.

(i) expected $ 9 $ order points $ 3 $, as $ # E[3] = # left ( mathbb {Z} / 3 mathbb {Z} times mathbb {Z} / 3 mathbb {Z} right) = $ 9. Because there is only $ 2 $ order points $ 3 $?

(ii) I calculated $ # E ( mathbb {Z} / 13 mathbb {Z}) = $ 9 (by brute force). So I understand the above **Corollary 6.4**, there must be some points of order $ 5 $, as $ p = 13 nmid 5 $. But the existence of such a point $ P in E[5]$ would imply the existence of some subgroup $ langle P rangle leq E ( mathbb {Z} / 13 mathbb {Z}) $ With order $ 5 $ as well. This can not happen, since the order of a subgroup must divide the order of the upper group. As $ 5 nmid 9 $, there can be no points of order $ 5 $.

What am I missing?