Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 5 x + 7 x^{2} )( 1 + 7 x^{2} )$ |
$1 - 5 x + 14 x^{2} - 35 x^{3} + 49 x^{4}$ | |
Frobenius angles: | $\pm0.106147807505$, $\pm0.5$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $2$ |
Isomorphism classes: | 10 |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $24$ | $2496$ | $111456$ | $5481216$ | $282929064$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $53$ | $324$ | $2281$ | $16833$ | $118622$ | $824799$ | $5764273$ | $40366188$ | $282541853$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+6x^5+2x$
- $y^2=4x^5+6x^4+5x^3+x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{2}}$.
Endomorphism algebra over $\F_{7}$The isogeny class factors as 1.7.af $\times$ 1.7.a 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 $\F_{7^{2}}$ is 1.49.al $\times$ 1.49.o. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.