Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 4 x + 13 x^{2} )( 1 - 6 x + 13 x^{2} )^{2}$ |
| $1 - 16 x + 123 x^{2} - 560 x^{3} + 1599 x^{4} - 2704 x^{5} + 2197 x^{6}$ | |
| Frobenius angles: | $\pm0.187167041811$, $\pm0.187167041811$, $\pm0.312832958189$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $640$ | $4608000$ | $11245402240$ | $23887872000000$ | $51482189428931200$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-2$ | $160$ | $2326$ | $29276$ | $373438$ | $4830880$ | $62753206$ | $815732156$ | $10604446078$ | $137857808800$ |
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.ag 2 $\times$ 1.13.ae 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.je 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$ |
- Endomorphism algebra over $\F_{13^{2}}$
The base change of $A$ to $\F_{13^{2}}$ is 1.169.ak 2 $\times$ 1.169.k. The endomorphism algebra for each factor is: - 1.169.ak 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 1.169.k : \(\Q(\sqrt{-1}) \).
Base change
This is a primitive isogeny class.