Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 4 x + 5 x^{2} )( 1 - 2 x + 5 x^{2} )$ |
$1 - 6 x + 18 x^{2} - 30 x^{3} + 25 x^{4}$ | |
Frobenius angles: | $\pm0.147583617650$, $\pm0.352416382350$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $1$ |
Isomorphism classes: | 6 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $8$ | $640$ | $18056$ | $409600$ | $9746888$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $26$ | $144$ | $654$ | $3120$ | $15626$ | $78624$ | $392734$ | $1956960$ | $9765626$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+3x^5+2x^4+2x^3+2x^2+3x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{4}}$.
Endomorphism algebra over $\F_{5}$The isogeny class factors as 1.5.ae $\times$ 1.5.ac 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 $\F_{5^{4}}$ is 1.625.o 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
- Endomorphism algebra over $\F_{5^{2}}$
The base change of $A$ to $\F_{5^{2}}$ is 1.25.ag $\times$ 1.25.g. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.