Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 4 x + 7 x^{2} )( 1 + 7 x^{2} )$ |
| $1 - 4 x + 14 x^{2} - 28 x^{3} + 49 x^{4}$ | |
| Frobenius angles: | $\pm0.227185525829$, $\pm0.5$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $6$ |
| Isomorphism classes: | 30 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $32$ | $3072$ | $125216$ | $5750784$ | $286475552$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $62$ | $364$ | $2398$ | $17044$ | $118622$ | $823036$ | $5755966$ | $40341028$ | $282486782$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=5x^6+5x^5+3x^4+x^3+3x^2+5x+5$
- $y^2=3x^5+2x^4+3x^3+2x^2+3x$
- $y^2=x^6+6x^5+5x^4+3x^3+x^2+x+3$
- $y^2=5x^5+4x^4+5x^3+2x^2+3x$
- $y^2=5x^6+4x^5+x^3+4x+5$
- $y^2=6x^6+x^5+4x^4+x^3+6x^2+4x+2$
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.ae $\times$ 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.ac $\times$ 1.49.o. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.