Invariants
Base field: | $\F_{13}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 7 x + 13 x^{2} )^{3}$ |
$1 - 21 x + 186 x^{2} - 889 x^{3} + 2418 x^{4} - 3549 x^{5} + 2197 x^{6}$ | |
Frobenius angles: | $\pm0.0772104791556$, $\pm0.0772104791556$, $\pm0.0772104791556$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $343$ | $3176523$ | $9636401152$ | $22836204908811$ | $51009911926504363$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-7$ | $101$ | $1988$ | $27989$ | $370013$ | $4825292$ | $62754545$ | $815792645$ | $10604854484$ | $137860172621$ |
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 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This is a primitive isogeny class.