Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $4$ |
| L-polynomial: | $( 1 - x - 4 x^{2} - 5 x^{3} + 25 x^{4} )^{2}$ |
| $1 - 2 x - 7 x^{2} - 2 x^{3} + 76 x^{4} - 10 x^{5} - 175 x^{6} - 250 x^{7} + 625 x^{8}$ | |
| Frobenius angles: | $\pm0.0948835201023$, $\pm0.0948835201023$, $\pm0.761550186769$, $\pm0.761550186769$ |
| Angle rank: | $1$ (numerical) |
| 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: | $4$ |
| Slopes: | $[0, 0, 0, 0, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $256$ | $200704$ | $157351936$ | $168401895424$ | $89434395064576$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $8$ | $70$ | $688$ | $2924$ | $15842$ | $79244$ | $390048$ | $1963150$ | $9773528$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains no Jacobian of a hyperelliptic curve, but it is unknown whether it contains a Jacobian of a non-hyperelliptic curve.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{3}}$.
Endomorphism algebra over $\F_{5}$| The isogeny class factors as 2.5.ab_ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}, \sqrt{-19})\)$)$ |
| The base change of $A$ to $\F_{5^{3}}$ is 1.125.ao 4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-19}) \)$)$ |
Base change
This is a primitive isogeny class.