Invariants
Base field: | $\F_{5}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 + 5 x^{2} )( 1 - 4 x + 5 x^{2} )^{2}$ |
$1 - 8 x + 31 x^{2} - 80 x^{3} + 155 x^{4} - 200 x^{5} + 125 x^{6}$ | |
Frobenius angles: | $\pm0.147583617650$, $\pm0.147583617650$, $\pm0.5$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $0$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $24$ | $14400$ | $1875384$ | $235929600$ | $32050265304$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-2$ | $24$ | $118$ | $604$ | $3278$ | $16344$ | $79238$ | $391484$ | $1955998$ | $9772824$ |
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 $\F_{5^{2}}$.
Endomorphism algebra over $\F_{5}$The isogeny class factors as 1.5.ae 2 $\times$ 1.5.a 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 $\F_{5^{2}}$ is 1.25.ag 2 $\times$ 1.25.k. The endomorphism algebra for each factor is:
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Base change
This is a primitive isogeny class.