Invariants
Base field: | $\F_{13}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 7 x + 13 x^{2} )( 1 - 5 x + 13 x^{2} )^{2}$ |
$1 - 17 x + 134 x^{2} - 617 x^{3} + 1742 x^{4} - 2873 x^{5} + 2197 x^{6}$ | |
Frobenius angles: | $\pm0.0772104791556$, $\pm0.256122854178$, $\pm0.256122854178$ |
Angle rank: | $1$ (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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $567$ | $4298427$ | $10946057472$ | $23694103094571$ | $51341099522009787$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-3$ | $149$ | $2268$ | $29045$ | $372417$ | $4825292$ | $62725317$ | $815638469$ | $10604381004$ | $137859336029$ |
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 $\F_{13^{6}}$.
Endomorphism algebra over $\F_{13}$The isogeny class factors as 1.13.ah $\times$ 1.13.af 2 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 $\F_{13^{6}}$ is 1.4826809.atm 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$ |
- Endomorphism algebra over $\F_{13^{2}}$
The base change of $A$ to $\F_{13^{2}}$ is 1.169.ax $\times$ 1.169.b 2 . The endomorphism algebra for each factor is: - 1.169.ax : \(\Q(\sqrt{-3}) \).
- 1.169.b 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- Endomorphism algebra over $\F_{13^{3}}$
The base change of $A$ to $\F_{13^{3}}$ is 1.2197.acs $\times$ 1.2197.cs 2 . The endomorphism algebra for each factor is: - 1.2197.acs : \(\Q(\sqrt{-3}) \).
- 1.2197.cs 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
Base change
This is a primitive isogeny class.