Invariants
Base field: | $\F_{5^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 5 x )^{2}( 1 - 5 x + 25 x^{2} )$ |
$1 - 15 x + 100 x^{2} - 375 x^{3} + 625 x^{4}$ | |
Frobenius angles: | $0$, $0$, $\pm0.333333333333$ |
Angle rank: | $0$ (numerical) |
Jacobians: | $0$ |
This isogeny class is not simple, primitive, not ordinary, and supersingular. It is principally polarizable.
Newton polygon
This isogeny class is supersingular.
$p$-rank: | $0$ |
Slopes: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $336$ | $374976$ | $244109376$ | $152343749376$ | $95275917959376$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $11$ | $601$ | $15626$ | $390001$ | $9756251$ | $244078126$ | $6103281251$ | $152587500001$ | $3814697265626$ | $95367421875001$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{12}}$.
Endomorphism algebra over $\F_{5^{2}}$The isogeny class factors as 1.25.ak $\times$ 1.25.af 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 $\F_{5^{12}}$ is 1.244140625.abufy 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$. |
- Endomorphism algebra over $\F_{5^{4}}$
The base change of $A$ to $\F_{5^{4}}$ is 1.625.aby $\times$ 1.625.z. The endomorphism algebra for each factor is: - 1.625.aby : the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.
- 1.625.z : \(\Q(\sqrt{-3}) \).
- Endomorphism algebra over $\F_{5^{6}}$
The base change of $A$ to $\F_{5^{6}}$ is 1.15625.ajq $\times$ 1.15625.jq. The endomorphism algebra for each factor is: - 1.15625.ajq : the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.
- 1.15625.jq : the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$.
Base change
This is a primitive isogeny class.