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Let $q_0$ be a prime and $q$ = $q_0^n$. Let $a(F_q/F_{q_0})$ denote any integer which is trace of Frobenius over the field $F_q$ for some elliptic curve which can be defined over $F_{q_0}$.

It is stated in Mazur's paper (Rational isogenies of Primes degree, Inventiones mathematicae, 1978) that $a(F_3^{12}/F_3)$ = 658, -1358, +1458.

I can get only $\pm$ 1458 or $\pm$ 729 or 0 if i use the following thm.

(It might be a simple answer. But, I don't see that how 658, - 1358 occurs).

Theorem: Let $q_0$ be a prime and $q$ = $q_0^n$. Then there exists an elliptic curve $E$ dened over $F_q$ such that the trace of the Frobenius equals to $\beta$ ($\beta$ $\leq$ $\lfloor 2\sqrt{q} \rfloor$) if and only if one of the following cases occur:

(i) $q_0$ does not divide $\beta$,

(ii). $q$ is a square (i.e. $n$ is even) and

$\beta$ = $\pm 2 \sqrt{q}$ or

$\beta$ = $\pm \sqrt{q}$ ($q_0$ $\nequiv$ \not\equiv$1 (mod 3)) or$\beta$= 0 and ($q_0\nequiv$\not\equiv$ 1 (mod 4)),

(iii) $q$ is not a square (i.e. $n$ is odd) and

$\beta$ = 0 or = $\beta$ = $\pm$ $q_0^{n+1/2}$ and $q_0$ = 2,3.

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Let $q_0$ be a prime and $q$ = $q_0^n$. Let $a(F_q/F_{q_0})$ denote any integer which is trace of Frobenius over the field $F_q$ for some elliptic curve which can be defined over $F_{q_0}$.

It is stated in Mazur's paper (Rational isogenies of Primes degree, Inventiones mathematicae, 1978) that $a(F_3^{12}/F_3)$ = 658, -1358, +1458.

I can get only $\pm$ 1458 or $\pm$ 729 or 0 if i use the following thm.

(It might be a simple answer. But, I don't see that how 658, - 1358 occurs).

Theorem: Let $q_0$ be a prime and $q$ = $q_0^n$. Then there exists an elliptic curve $E$ dened over $F_q$ such that the trace of the Frobenius equals to $\beta$ ($\beta$ $\leq$ $\lfloor 2\sqrt{q} \rfloor$) if and only if one of the following cases occur:

1. $q_0$ does not divide $\beta$ \beta$, 2.$q$is a square (i.e.$n$is even) and$\beta$=$\pm 2 \sqrt{q}$or$\beta$=$\pm \sqrt{q}$($q_0\nequiv$1 (mod 3)) or$\beta$=0 \beta$= 0 and ($q_0$ $\nequiv$ 1 (mod 4))4)),

1. $q$ is not a square (i.e. $n$ is odd) and $\beta$ = 0 or = $\beta$ = $\pm$ $q_0^{n+1/2}$ and $q_0$ = 2,3.
Let $q_0$ be a prime and $q$ = $q_0^n$. Let $a(F_q/F_{q_0})$ denote any integer which is trace of Frobenius over the field $F_q$ for some elliptic curve which can be defined over $F_{q_0}$.
It is stated in Mazur's paper (Rational isogenies of Primes degree, Inventiones mathematicae, 1978) that $a(F_3^{12}/F_3)$ = 658, -1358, +1458.
I can get only $\pm$ 1458 or $\pm$ 729 or 0 if i use the following thm.
Theorem: Let $q_0$ be a prime and $q$ = $q_0^n$. Then there exists an elliptic curve $E$ dened over $F_q$ such that the trace of the Frobenius equals to $beta$ \beta$($\beta\leq\lfloor 2\sqrt{q} \rfloor$) if and only if one of the following cases occur: 1.$q_0$does not divide$\beta$2.$q$is a square (i.e.$n$is even) and$\beta$=$\pm 2 \sqrt{q}$or$\beta$=$\pm \sqrt{q}$($q_0\nequiv$1 (mod 3)) or$beta$=0 \beta$=0 and ($q_0$ $\nequiv$ 1 (mod 4))
1. $q$ is not a square (i.e. $n$ is odd) and $\beta$ = 0 or = $\beta$ = $\pm$ $q_0^{n+1/2}$ and $q_0$ = 2,3.