Invariants
Base field: | $\F_{5}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 3 x + 5 x^{2} )( 1 - 2 x + 5 x^{2} )( 1 - 4 x + 5 x^{2} )^{2}$ |
$1 - 13 x + 82 x^{2} - 323 x^{3} + 866 x^{4} - 1615 x^{5} + 2050 x^{6} - 1625 x^{7} + 625 x^{8}$ | |
Frobenius angles: | $\pm0.147583617650$, $\pm0.147583617650$, $\pm0.265942140215$, $\pm0.352416382350$ |
Angle rank: | $2$ (numerical) |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $48$ | $345600$ | $317207808$ | $176947200000$ | $99339951101808$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-7$ | $21$ | $158$ | $717$ | $3253$ | $15786$ | $78673$ | $392637$ | $1957478$ | $9769101$ |
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 $\F_{5^{4}}$.
Endomorphism algebra over $\F_{5}$The isogeny class factors as 1.5.ae 2 $\times$ 1.5.ad $\times$ 1.5.ac 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 $\F_{5^{4}}$ is 1.625.o 3 $\times$ 1.625.bx. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{5^{2}}$
The base change of $A$ to $\F_{5^{2}}$ is 1.25.ag 2 $\times$ 1.25.b $\times$ 1.25.g. The endomorphism algebra for each factor is: - 1.25.ag 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 1.25.b : \(\Q(\sqrt{-11}) \).
- 1.25.g : \(\Q(\sqrt{-1}) \).
Base change
This is a primitive isogeny class.