Invariants
Base field: | $\F_{7}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 5 x + 7 x^{2} )( 1 - 6 x + 18 x^{2} - 42 x^{3} + 49 x^{4} )$ |
$1 - 11 x + 55 x^{2} - 174 x^{3} + 385 x^{4} - 539 x^{5} + 343 x^{6}$ | |
Frobenius angles: | $\pm0.0461154155528$, $\pm0.106147807505$, $\pm0.453884584447$ |
Angle rank: | $2$ (numerical) |
Isomorphism classes: | 3 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $60$ | $90480$ | $36164880$ | $12804729600$ | $4652481286500$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-3$ | $39$ | $306$ | $2215$ | $16467$ | $117936$ | $825381$ | $5765231$ | $40350582$ | $282508239$ |
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 $\F_{7^{4}}$.
Endomorphism algebra over $\F_{7}$The isogeny class factors as 1.7.af $\times$ 2.7.ag_s 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 $\F_{7^{4}}$ is 1.2401.ade 2 $\times$ 1.2401.ax. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{7^{2}}$
The base change of $A$ to $\F_{7^{2}}$ is 1.49.al $\times$ 2.49.a_ade. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.