Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 7 x^{2} )( 1 + 4 x + 7 x^{2} )$ |
| $1 + 4 x + 14 x^{2} + 28 x^{3} + 49 x^{4}$ | |
| Frobenius angles: | $\pm0.5$, $\pm0.772814474171$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $6$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $96$ | $3072$ | $111456$ | $5750784$ | $278542176$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $62$ | $324$ | $2398$ | $16572$ | $118622$ | $824052$ | $5755966$ | $40366188$ | $282486782$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=4 x^6+4 x^5+x^4+5 x^3+x^2+4 x+4$
- $y^2=x^5+3 x^4+x^3+3 x^2+x$
- $y^2=3 x^6+4 x^5+x^4+2 x^3+3 x^2+3 x+2$
- $y^2=4 x^5+6 x^4+4 x^3+3 x^2+x$
- $y^2=4 x^6+6 x^5+5 x^3+6 x+4$
- $y^2=4 x^6+3 x^5+5 x^4+3 x^3+4 x^2+5 x+6$
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.a $\times$ 1.7.e 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.ac $\times$ 1.49.o. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.