Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 4 x + 7 x^{2} )( 1 - 2 x + 7 x^{2} )$ |
$1 - 6 x + 22 x^{2} - 42 x^{3} + 49 x^{4}$ | |
Frobenius angles: | $\pm0.227185525829$, $\pm0.376624142786$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $4$ |
Isomorphism classes: | 12 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $24$ | $2880$ | $137592$ | $5990400$ | $282350904$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $2$ | $58$ | $398$ | $2494$ | $16802$ | $117466$ | $823790$ | $5765566$ | $40345346$ | $282428218$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+3x^5+3x^4+4x^3+5x^2+6x+6$
- $y^2=6x^6+x^4+x^3+x^2+6$
- $y^2=6x^6+3x^5+x^4+x^3+4x^2+6x+6$
- $y^2=5x^6+3x^5+6x^4+4x^3+6x^2+3x+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7}$.
Endomorphism algebra over $\F_{7}$The isogeny class factors as 1.7.ae $\times$ 1.7.ac 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.