Invariants
Base field: | $\F_{5}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 4 x + 5 x^{2} )( 1 - 3 x + 5 x^{2} )( 1 - 4 x + 8 x^{2} - 20 x^{3} + 25 x^{4} )$ |
$1 - 11 x + 58 x^{2} - 199 x^{3} + 506 x^{4} - 995 x^{5} + 1450 x^{6} - 1375 x^{7} + 625 x^{8}$ | |
Frobenius angles: | $\pm0.0320471084245$, $\pm0.147583617650$, $\pm0.265942140215$, $\pm0.532047108424$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $60$ | $313200$ | $219424320$ | $145324800000$ | $98138950212300$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-5$ | $21$ | $112$ | $597$ | $3215$ | $15786$ | $77387$ | $388797$ | $1954120$ | $9769101$ |
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 $\times$ 1.5.ad $\times$ 2.5.ae_i 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.abu 2 $\times$ 1.625.o $\times$ 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 $\times$ 1.25.b $\times$ 2.25.a_abu. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.