Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 5 x + 13 x^{2} )( 1 - 7 x + 13 x^{2} )^{2}$ |
| $1 - 19 x + 158 x^{2} - 739 x^{3} + 2054 x^{4} - 3211 x^{5} + 2197 x^{6}$ | |
| Frobenius angles: | $\pm0.0772104791556$, $\pm0.0772104791556$, $\pm0.256122854178$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $441$ | $3695139$ | $10270374912$ | $23261199311259$ | $51175237809194541$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-5$ | $125$ | $2128$ | $28517$ | $371215$ | $4825292$ | $62739931$ | $815715557$ | $10604617744$ | $137859754325$ |
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 2 $\times$ 1.13.af 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 2 $\times$ 1.169.b. The endomorphism algebra for each factor is: - 1.169.ax 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 1.169.b : \(\Q(\sqrt{-3}) \).
- Endomorphism algebra over $\F_{13^{3}}$
The base change of $A$ to $\F_{13^{3}}$ is 1.2197.acs 2 $\times$ 1.2197.cs. The endomorphism algebra for each factor is: - 1.2197.acs 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 1.2197.cs : \(\Q(\sqrt{-3}) \).
Base change
This is a primitive isogeny class.