Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 4 x + 5 x^{2} )( 1 + 4 x + 5 x^{2} )$ |
| $1 - 6 x^{2} + 25 x^{4}$ | |
| Frobenius angles: | $\pm0.147583617650$, $\pm0.852416382350$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $2$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $20$ | $400$ | $15860$ | $409600$ | $9766100$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $6$ | $14$ | $126$ | $654$ | $3126$ | $16094$ | $78126$ | $392734$ | $1953126$ | $9766574$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 2 curves (of which all are hyperelliptic):
- $y^2=x^6+4 x^5+2 x^4+3 x^2+x+4$
- $y^2=x^6+x^5+x^3+4 x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{2}}$.
Endomorphism algebra over $\F_{5}$| The isogeny class factors as 1.5.ae $\times$ 1.5.e 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 $\F_{5^{2}}$ is 1.25.ag 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
Base change
This is a primitive isogeny class.