# elliptic curve (y ^ 2 = x ^ 3 + 3x +8) (mod 13) – torsion groups

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 $$# E stackrel {?} {=} 2$$ order points $$3$$ Y $$# E stackrel {?} {=} 6$$ Points of order 9.

(i) expected $$9$$ order points $$3$$, as $$# E = # left ( mathbb {Z} / 3 mathbb {Z} times mathbb {Z} / 3 mathbb {Z} right) = 9$$. 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 $$P in E$$ 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?