Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 2 x + 5 x^{2} )( 1 - 4 x + 5 x^{2} )^{2}$ |
| $1 - 10 x + 47 x^{2} - 132 x^{3} + 235 x^{4} - 250 x^{5} + 125 x^{6}$ | |
| Frobenius angles: | $\pm0.147583617650$, $\pm0.147583617650$, $\pm0.352416382350$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $0$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $16$ | $12800$ | $2202832$ | $262144000$ | $31209535376$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-4$ | $20$ | $140$ | $668$ | $3196$ | $15860$ | $79180$ | $393788$ | $1958396$ | $9766100$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
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 2 $\times$ 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 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$ |
- Endomorphism algebra over $\F_{5^{2}}$
The base change of $A$ to $\F_{5^{2}}$ is 1.25.ag 2 $\times$ 1.25.g. The endomorphism algebra for each factor is: - 1.25.ag 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 1.25.g : \(\Q(\sqrt{-1}) \).
Base change
This is a primitive isogeny class.