Invariants
Base field: | $\F_{13}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 7 x + 13 x^{2} )( 1 - 8 x + 36 x^{2} - 104 x^{3} + 169 x^{4} )$ |
$1 - 15 x + 105 x^{2} - 460 x^{3} + 1365 x^{4} - 2535 x^{5} + 2197 x^{6}$ | |
Frobenius angles: | $\pm0.0772104791556$, $\pm0.147614849952$, $\pm0.431019279425$ |
Angle rank: | $3$ (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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $658$ | $4394124$ | $10465874272$ | $23043647504304$ | $51092959908644458$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-1$ | $155$ | $2168$ | $28247$ | $370619$ | $4831592$ | $62782103$ | $815808767$ | $10604532104$ | $137858690675$ |
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 $\times$ 2.13.ai_bk 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.