Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 5 x + 7 x^{2} )( 1 - x + 7 x^{2} )$ |
$1 - 6 x + 19 x^{2} - 42 x^{3} + 49 x^{4}$ | |
Frobenius angles: | $\pm0.106147807505$, $\pm0.439481140838$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $4$ |
Isomorphism classes: | 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $21$ | $2457$ | $117936$ | $5545449$ | $279377301$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $2$ | $52$ | $344$ | $2308$ | $16622$ | $118222$ | $826562$ | $5768836$ | $40353608$ | $282497332$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+x^3+3$
- $y^2=3x^6+x^5+2x^4+2x^2+x+3$
- $y^2=5x^6+6x^5+x^4+2x+5$
- $y^2=x^6+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{6}}$.
Endomorphism algebra over $\F_{7}$The isogeny class factors as 1.7.af $\times$ 1.7.ab 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 $\F_{7^{6}}$ is 1.117649.la 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
- Endomorphism algebra over $\F_{7^{2}}$
The base change of $A$ to $\F_{7^{2}}$ is 1.49.al $\times$ 1.49.n. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{7^{3}}$
The base change of $A$ to $\F_{7^{3}}$ is 1.343.au $\times$ 1.343.u. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.