Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 4 x + 7 x^{2} )( 1 + x + 7 x^{2} )$ |
$1 - 3 x + 10 x^{2} - 21 x^{3} + 49 x^{4}$ | |
Frobenius angles: | $\pm0.227185525829$, $\pm0.560518859162$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $6$ |
Isomorphism classes: | 28 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $36$ | $3024$ | $117936$ | $5818176$ | $290071836$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $61$ | $344$ | $2425$ | $17255$ | $118222$ | $821273$ | $5760529$ | $40353608$ | $282442261$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+6x^5+2x^4+3x^3+x^2+4x+6$
- $y^2=4x^6+2x^5+4x^4+3x^3+3x^2+2x+1$
- $y^2=5x^5+3x^4+x^3+4x^2+2x+6$
- $y^2=3x^6+2x^5+6x^4+6x^3+x^2+3x+3$
- $y^2=x^6+x^3+5$
- $y^2=3x^6+5x^5+3x^4+5x^3+6x^2+4x+5$
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.ae $\times$ 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 $\F_{7^{6}}$ 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.ac $\times$ 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.