Invariants
Base field: | $\F_{7}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 5 x + 7 x^{2} )( 1 - 3 x + 7 x^{2} )$ |
$1 - 8 x + 29 x^{2} - 56 x^{3} + 49 x^{4}$ | |
Frobenius angles: | $\pm0.106147807505$, $\pm0.308124534521$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $1$ |
Isomorphism classes: | 2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $15$ | $2145$ | $123120$ | $5888025$ | $282373575$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $44$ | $360$ | $2452$ | $16800$ | $117326$ | $823200$ | $5768548$ | $40375800$ | $282540764$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+x^5+6x^4+2x^3+3x^2+2x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7}$.
Endomorphism algebra over $\F_{7}$The isogeny class factors as 1.7.af $\times$ 1.7.ad 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.