Invariants
Base field: | $\F_{5}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 4 x + 5 x^{2} )^{2}( 1 - 4 x + 8 x^{2} - 20 x^{3} + 25 x^{4} )$ |
$1 - 12 x + 66 x^{2} - 228 x^{3} + 578 x^{4} - 1140 x^{5} + 1650 x^{6} - 1500 x^{7} + 625 x^{8}$ | |
Frobenius angles: | $\pm0.0320471084245$, $\pm0.147583617650$, $\pm0.147583617650$, $\pm0.532047108424$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $40$ | $232000$ | $185901160$ | $137789440000$ | $98724762356200$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-6$ | $14$ | $90$ | $562$ | $3234$ | $16094$ | $78450$ | $391002$ | $1956474$ | $9766574$ |
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 $\times$ 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 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 2 $\times$ 2.25.a_abu. The endomorphism algebra for each factor is: - 1.25.ag 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 2.25.a_abu : \(\Q(i, \sqrt{6})\).
Base change
This is a primitive isogeny class.