Invariants
Base field: | $\F_{5}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 4 x + 5 x^{2} )( 1 - 3 x + 5 x^{2} )( 1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4} )$ |
$1 - 13 x + 81 x^{2} - 316 x^{3} + 844 x^{4} - 1580 x^{5} + 2025 x^{6} - 1625 x^{7} + 625 x^{8}$ | |
Frobenius angles: | $\pm0.0512862249088$, $\pm0.147583617650$, $\pm0.265942140215$, $\pm0.384619558242$ |
Angle rank: | $3$ (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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $42$ | $298620$ | $272022912$ | $156954672000$ | $93754335863202$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-7$ | $19$ | $140$ | $643$ | $3073$ | $15502$ | $78253$ | $391395$ | $1953644$ | $9763819$ |
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 $\F_{5^{6}}$.
Endomorphism algebra over $\F_{5}$The isogeny class factors as 1.5.ae $\times$ 1.5.ad $\times$ 2.5.ag_r 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 $\F_{5^{6}}$ is 1.15625.afm 2 $\times$ 1.15625.acw $\times$ 1.15625.ja. 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 $\times$ 1.25.b $\times$ 2.25.ac_av. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{5^{3}}$
The base change of $A$ to $\F_{5^{3}}$ is 1.125.ae $\times$ 1.125.s $\times$ 2.125.a_afm. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.