Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $1 - x - 4 x^{2} - 5 x^{3} + 25 x^{4}$ |
Frobenius angles: | $\pm0.0948835201023$, $\pm0.761550186769$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-3}, \sqrt{-19})\) |
Galois group: | $C_2^2$ |
Jacobians: | $1$ |
Isomorphism classes: | 3 |
This isogeny class is simple but not geometrically 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $16$ | $448$ | $12544$ | $410368$ | $9456976$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $17$ | $98$ | $657$ | $3025$ | $15734$ | $78685$ | $390337$ | $1958138$ | $9769577$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=4x^5+2x^4+x^3+4x^2+4x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{3}}$.
Endomorphism algebra over $\F_{5}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-19})\). |
The base change of $A$ to $\F_{5^{3}}$ is 1.125.ao 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$ |
Base change
This is a primitive isogeny class.