Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 5 x + 7 x^{2} )( 1 + x + 7 x^{2} )$ |
$1 - 4 x + 9 x^{2} - 28 x^{3} + 49 x^{4}$ | |
Frobenius angles: | $\pm0.106147807505$, $\pm0.560518859162$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $6$ |
Isomorphism classes: | 14 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $27$ | $2457$ | $104976$ | $5545449$ | $286480827$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $52$ | $304$ | $2308$ | $17044$ | $118222$ | $823036$ | $5768836$ | $40378768$ | $282497332$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+4x^5+3x^3+x+5$
- $y^2=3x^6+5x^4+3x^3+5x^2+3$
- $y^2=3x^6+2x^5+2x^3+5x^2+2x+3$
- $y^2=x^6+4$
- $y^2=3x^6+4x^5+x^4+x^2+x+6$
- $y^2=3x^6+2x^5+6x^4+3x^3+2x^2+6x+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{3}}$.
Endomorphism algebra over $\F_{7}$The isogeny class factors as 1.7.af $\times$ 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: |
The base change of $A$ to $\F_{7^{3}}$ is 1.343.au 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This is a primitive isogeny class.