Invariants
This isogeny class is simple and geometrically simple,
primitive,
not ordinary,
and not supersingular.
It is principally polarizable and
contains a Jacobian.
Point counts
Point counts of the abelian variety
| $r$ |
$1$ |
$2$ |
$3$ |
$4$ |
$5$ |
|---|
| $A(\F_{q^r})$ |
$19$ |
$361$ |
$4009$ |
$130321$ |
$1050529$ |
Point counts of the curve
| $r$ |
$1$ |
$2$ |
$3$ |
$4$ |
$5$ |
$6$ |
$7$ |
$8$ |
$9$ |
$10$ |
|---|
| $C(\F_{q^r})$ |
$3$ |
$5$ |
$9$ |
$25$ |
$33$ |
$65$ |
$129$ |
$369$ |
$513$ |
$1025$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is not hyperelliptic):
- $x y+t^2=x^3+x^2 y+y^3+y^2 z+z^3+x^2 t=0$
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2}$
Endomorphism algebra over $\overline{\F}_{2}$
| The base change of $A$ to $\F_{2^{4}}$ is the simple isogeny class 4.16.i_dk_qa_dlg and its endomorphism algebra is the division algebra of dimension 16 over \(\Q(\sqrt{-15}) \) with the following ramification data at primes above $2$, and unramified at all archimedean places: |
| $v$ | ($ 2 $,\( \pi \)) | ($ 2 $,\( \pi + 1 \)) | | $\operatorname{inv}_v$ | $1/4$ | $3/4$ |
where $\pi$ is a root of $x^{2} - x + 4$.
|
Remainder of endomorphism lattice by field
- Endomorphism algebra over $\F_{2^{2}}$
| The base change of $A$ to $\F_{2^{2}}$ is the simple isogeny class 4.4.a_e_a_bk and its endomorphism algebra is the quaternion algebra over \(\Q(\sqrt{6}, \sqrt{-10})\) with the following ramification data at primes above $2$, and unramified at all archimedean places: |
| $v$ | ($ 2 $,\( \frac{1}{2} \pi^{2} + \pi + 1 \)) | ($ 2 $,\( \frac{7}{4} \pi^{3} + \frac{1}{2} \pi^{2} + \frac{1}{2} \pi \)) | | $\operatorname{inv}_v$ | $1/2$ | $1/2$ |
where $\pi$ is a root of $x^{4} + 2x^{2} + 16$.
|
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|
| 4.2.a_a_a_ac | $8$ | (not in LMFDB) |
