Let $A$ be an abelian variety over $\mathbb{F}\_p$.
Then of course for every natural number $i$, we have that $\# A(\mathbb{F}\_{p^i})$ divides $\# A(\mathbb{F}\_{p^{i+1}})$.


But MAGMA says this is false:
Here is my code:



```
P<x> := PolynomialRing((FiniteField(3)));
J := Jacobian(HyperellipticCurve(x^6 - 2 * x^5 + x^4 - 2 * x^3 + 6 * x^2 - 4 * x + 1));
for j in [1..10] do;
    Order(BaseChange(J, FiniteField(3, j)));
end for;

```

And the result is:


19
57
1444
5529
59299
467856
4976347
43264425
394975876
3458495577


What is wrong?



The orders of $\mathbb{F}_{p^n}$- rational points of a fixed abelian variety and MAGMA computation
$\mathbf F\_{p^i}$ is only a subfield of $\mathbf F\_{p^j}$ when $i |j$ so you only have the divisibility of group orders for $i|j$ not for $j = i+1$.


