Invariants
Base field: | $\F_{5}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 4 x + 5 x^{2} )( 1 - 2 x + 5 x^{2} )( 1 - x + 5 x^{2} )$ |
$1 - 7 x + 29 x^{2} - 78 x^{3} + 145 x^{4} - 175 x^{5} + 125 x^{6}$ | |
Frobenius angles: | $\pm0.147583617650$, $\pm0.352416382350$, $\pm0.428216853436$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $0$ |
Isomorphism classes: | 20 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $40$ | $22400$ | $2527840$ | $243712000$ | $29484336200$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-1$ | $35$ | $158$ | $623$ | $3019$ | $15680$ | $79183$ | $393023$ | $1954454$ | $9761675$ |
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 $\times$ 1.5.ac $\times$ 1.5.ab 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.abf $\times$ 1.625.o 2 . 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.g $\times$ 1.25.j. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.