Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 3 x + 7 x^{2} )( 1 + x + 7 x^{2} )$ |
| $1 - 2 x + 11 x^{2} - 14 x^{3} + 49 x^{4}$ | |
| Frobenius angles: | $\pm0.308124534521$, $\pm0.560518859162$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $8$ |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $45$ | $3465$ | $123120$ | $5769225$ | $285493725$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $68$ | $360$ | $2404$ | $16986$ | $117326$ | $820182$ | $5764036$ | $40375800$ | $282496868$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=4 x^6+2 x^4+2 x^3+x^2+2 x+6$
- $y^2=x^6+2 x^5+6 x^4+x^3+4 x^2+6 x+4$
- $y^2=5 x^6+3 x^5+3 x^4+2 x^3+3 x^2+3 x+5$
- $y^2=x^6+x^4+2 x^3+x^2+1$
- $y^2=x^6+4 x^4+2 x^3+3 x+2$
- $y^2=2 x^6+3 x^4+2 x^3+3 x^2+2$
- $y^2=5 x^6+2 x^4+3 x^3+2 x^2+5$
- $y^2=3 x^6+6 x^4+5 x^3+2 x^2+2 x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7}$.
Endomorphism algebra over $\F_{7}$| The isogeny class factors as 1.7.ad $\times$ 1.7.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.