Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 4 x + 5 x^{2} )( 1 + 2 x + 5 x^{2} )$ |
| $1 - 2 x + 2 x^{2} - 10 x^{3} + 25 x^{4}$ | |
| Frobenius angles: | $\pm0.147583617650$, $\pm0.647583617650$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $5$ |
| Isomorphism classes: | 20 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $16$ | $640$ | $12688$ | $409600$ | $10272016$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $26$ | $100$ | $654$ | $3284$ | $15626$ | $78740$ | $392734$ | $1952164$ | $9765626$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=4x^5+3x^4+3$
- $y^2=3x^6+4x^5+3x^4+3x^3+3x^2+4x+3$
- $y^2=2x^6+4x^5+3x^4+3x^2+x+2$
- $y^2=x^6+2x^5+2x^4+4x^3+4x^2+2x+3$
- $y^2=3x^5+2x^4+2x^3+4x^2+3x+2$
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.c 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.