Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 7 x + 13 x^{2} )( 1 - 6 x + 13 x^{2} )( 1 - 4 x + 13 x^{2} )$ |
| $1 - 17 x + 133 x^{2} - 610 x^{3} + 1729 x^{4} - 2873 x^{5} + 2197 x^{6}$ | |
| Frobenius angles: | $\pm0.0772104791556$, $\pm0.187167041811$, $\pm0.312832958189$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $560$ | $4233600$ | $10798833920$ | $23532042240000$ | $51258148307970800$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-3$ | $147$ | $2238$ | $28847$ | $371817$ | $4826304$ | $62746317$ | $815752319$ | $10604677254$ | $137859052107$ |
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 $\times$ 1.13.ag $\times$ 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.ahj $\times$ 1.28561.je 2 . 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 $\times$ 1.169.ak $\times$ 1.169.k. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.