Invariants
Base field: | $\F_{13}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 4 x + 13 x^{2} )( 1 - 7 x + 13 x^{2} )^{2}$ |
$1 - 18 x + 144 x^{2} - 664 x^{3} + 1872 x^{4} - 3042 x^{5} + 2197 x^{6}$ | |
Frobenius angles: | $\pm0.0772104791556$, $\pm0.0772104791556$, $\pm0.312832958189$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $490$ | $3889620$ | $10369999360$ | $23181512860800$ | $51035082173203450$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-4$ | $134$ | $2150$ | $28418$ | $370196$ | $4821728$ | $62739428$ | $815772482$ | $10604908430$ | $137860295414$ |
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.ah 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:
|
Base change
This is a primitive isogeny class.