Invariants
Base field: | $\F_{5}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 3 x + 5 x^{2} )^{2}( 1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4} )$ |
$1 - 12 x + 72 x^{2} - 276 x^{3} + 733 x^{4} - 1380 x^{5} + 1800 x^{6} - 1500 x^{7} + 625 x^{8}$ | |
Frobenius angles: | $\pm0.0512862249088$, $\pm0.265942140215$, $\pm0.265942140215$, $\pm0.384619558242$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $63$ | $403137$ | $321076224$ | $165538130625$ | $93198017193183$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-6$ | $26$ | $162$ | $678$ | $3054$ | $15194$ | $77190$ | $389190$ | $1951290$ | $9766346$ |
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.ad 2 $\times$ 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 2 . The endomorphism algebra for each factor is:
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- Endomorphism algebra over $\F_{5^{2}}$
The base change of $A$ to $\F_{5^{2}}$ is 1.25.b 2 $\times$ 2.25.ac_av. The endomorphism algebra for each factor is: - 1.25.b 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
- 2.25.ac_av : \(\Q(\sqrt{2}, \sqrt{-3})\).
- Endomorphism algebra over $\F_{5^{3}}$
The base change of $A$ to $\F_{5^{3}}$ is 1.125.s 2 $\times$ 2.125.a_afm. The endomorphism algebra for each factor is: - 1.125.s 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$
- 2.125.a_afm : \(\Q(\sqrt{2}, \sqrt{-3})\).
Base change
This is a primitive isogeny class.