Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $4$ |
| L-polynomial: | $( 1 + 4 x^{2} + 25 x^{4} )^{2}$ |
| $1 + 8 x^{2} + 66 x^{4} + 200 x^{6} + 625 x^{8}$ | |
| Frobenius angles: | $\pm0.315494940217$, $\pm0.315494940217$, $\pm0.684505059783$, $\pm0.684505059783$ |
| Angle rank: | $1$ (numerical) |
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: | $4$ |
| Slopes: | $[0, 0, 0, 0, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $900$ | $810000$ | $236852100$ | $189747360000$ | $95475372322500$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $42$ | $126$ | $762$ | $3126$ | $14682$ | $78126$ | $391002$ | $1953126$ | $9787722$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^{10}+2 x^6+x^4+3$
- $y^2=x^{10}+x^8+2 x^7+4 x^6+x^5+x^4+2$
- $y^2=2 x^{10}+2 x^8+4 x^7+3 x^6+2 x^5+2 x^4+4$
- $y^2=x^{10}+3 x^9+4 x^8+2 x^6+x^4+3 x^2+2 x+3$
- $y^2=x^{10}+3 x^6+2 x^4+4$
- $y^2=2 x^{10}+x^6+4 x^4+3$
- $y^2=x^9+x^7+2 x^6+2 x^5+4 x^3+3 x^2+4$
- $y^2=2 x^9+2 x^7+4 x^6+4 x^5+3 x^3+x^2+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{2}}$.
Endomorphism algebra over $\F_{5}$| The isogeny class factors as 2.5.a_e 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{6}, \sqrt{-14})\)$)$ |
| The base change of $A$ to $\F_{5^{2}}$ is 1.25.e 4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-21}) \)$)$ |
Base change
This is a primitive isogeny class.