Invariants
Base field: | $\F_{13}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 32 x^{2} - 104 x^{3} + 169 x^{4} )$ |
$1 - 15 x + 101 x^{2} - 432 x^{3} + 1313 x^{4} - 2535 x^{5} + 2197 x^{6}$ | |
Frobenius angles: | $\pm0.0370621216586$, $\pm0.0772104791556$, $\pm0.462937878341$ |
Angle rank: | $2$ (numerical) |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $630$ | $4154220$ | $10012857120$ | $22657863639600$ | $50895254892840750$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-1$ | $147$ | $2072$ | $27767$ | $369179$ | $4826304$ | $62751191$ | $815683199$ | $10604336456$ | $137859052107$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{4}}$.
Endomorphism algebra over $\F_{13}$The isogeny class factors as 1.13.ah $\times$ 2.13.ai_bg and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
The base change of $A$ to $\F_{13^{4}}$ is 1.28561.alq 2 $\times$ 1.28561.ahj. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{13^{2}}$
The base change of $A$ to $\F_{13^{2}}$ is 1.169.ax $\times$ 2.169.a_alq. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.