Invariants
This isogeny class is simple and geometrically simple,
not primitive,
not ordinary,
and not supersingular.
It is principally polarizable.
Point counts
Point counts of the abelian variety
| $r$ |
$1$ |
$2$ |
$3$ |
$4$ |
$5$ |
|---|
| $A(\F_{q^r})$ |
$529$ |
$279841$ |
$16200625$ |
$4097152081$ |
$1095691843009$ |
Point counts of the (virtual) curve
| $r$ |
$1$ |
$2$ |
$3$ |
$4$ |
$5$ |
$6$ |
$7$ |
$8$ |
$9$ |
$10$ |
|---|
| $C(\F_{q^r})$ |
$5$ |
$41$ |
$65$ |
$241$ |
$1025$ |
$3809$ |
$16385$ |
$67521$ |
$262145$ |
$1041281$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but it is unknown whether it contains a Jacobian.
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2^{2}}$
| The endomorphism algebra of this simple isogeny class is the quaternion algebra over \(\Q(\sqrt{2}, \sqrt{-7})\) with the following ramification data at primes above $2$, and unramified at all archimedean places: |
| $v$ | ($ 2 $,\( \frac{3}{2} \pi^{2} + \pi + 1 \)) | ($ 2 $,\( \frac{1}{4} \pi^{3} + \frac{1}{2} \pi^{2} + \frac{3}{2} \pi \)) | | $\operatorname{inv}_v$ | $1/2$ | $1/2$ |
where $\pi$ is a root of $x^{4} + 6x^{2} + 16$.
|
Endomorphism algebra over $\overline{\F}_{2^{2}}$
| The base change of $A$ to $\F_{2^{4}}$ is the simple isogeny class 4.16.y_ku_czo_oku and its endomorphism algebra is the division algebra of dimension 16 over \(\Q(\sqrt{-7}) \) with the following ramification data at primes above $2$, and unramified at all archimedean places: |
| $v$ | ($ 2 $,\( \pi + 1 \)) | ($ 2 $,\( \pi \)) | | $\operatorname{inv}_v$ | $1/4$ | $3/4$ |
where $\pi$ is a root of $x^{2} - x + 2$.
|
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{2}}$.
Twists
