Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - x + 5 x^{2} )^{2}$ |
| $1 - 2 x + 11 x^{2} - 10 x^{3} + 25 x^{4}$ | |
| Frobenius angles: | $\pm0.428216853436$, $\pm0.428216853436$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $1$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $25$ | $1225$ | $19600$ | $354025$ | $9150625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $44$ | $154$ | $564$ | $2924$ | $15734$ | $79244$ | $391204$ | $1948114$ | $9757724$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobian of 1 curve (which is hyperelliptic):
- $y^2=4x^6+x^5+x^4+x^3+x^2+x+4$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5}$.
Endomorphism algebra over $\F_{5}$| The isogeny class factors as 1.5.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$ |
Base change
This is a primitive isogeny class.