Invariants
Base field: | $\F_{7}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 5 x + 7 x^{2} )( 1 - 4 x + 8 x^{2} - 28 x^{3} + 49 x^{4} )$ |
$1 - 9 x + 35 x^{2} - 96 x^{3} + 245 x^{4} - 441 x^{5} + 343 x^{6}$ | |
Frobenius angles: | $\pm0.0704914820143$, $\pm0.106147807505$, $\pm0.570491482014$ |
Angle rank: | $2$ (numerical) |
Isomorphism classes: | 6 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $78$ | $91260$ | $32760936$ | $13026452400$ | $4787838907098$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-1$ | $39$ | $272$ | $2255$ | $16949$ | $117936$ | $822947$ | $5770991$ | $40382984$ | $282508239$ |
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.ae_i 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.ack 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_ack. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.