Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 5 x + 13 x^{2} )( 1 - 6 x + 13 x^{2} )^{2}$ |
| $1 - 17 x + 135 x^{2} - 622 x^{3} + 1755 x^{4} - 2873 x^{5} + 2197 x^{6}$ | |
| Frobenius angles: | $\pm0.187167041811$, $\pm0.187167041811$, $\pm0.256122854178$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $576$ | $4377600$ | $11137367808$ | $23969986560000$ | $51623572937967936$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-3$ | $151$ | $2304$ | $29375$ | $374457$ | $4834444$ | $62753709$ | $815675231$ | $10604155392$ | $137857267711$ |
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 $\F_{13}$.
Endomorphism algebra over $\F_{13}$The isogeny class factors as 1.13.ag 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:
|
Base change
This is a primitive isogeny class.