Invariants
Base field: | $\F_{5}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 4 x + 5 x^{2} )( 1 - 2 x + 5 x^{2} )( 1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4} )$ |
$1 - 12 x + 71 x^{2} - 270 x^{3} + 716 x^{4} - 1350 x^{5} + 1775 x^{6} - 1500 x^{7} + 625 x^{8}$ | |
Frobenius angles: | $\pm0.0512862249088$, $\pm0.147583617650$, $\pm0.352416382350$, $\pm0.384619558242$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $56$ | $353920$ | $279579104$ | $148816281600$ | $89660131438136$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-6$ | $24$ | $144$ | $608$ | $2934$ | $15342$ | $78702$ | $393600$ | $1956960$ | $9760344$ |
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 $\F_{5^{12}}$.
Endomorphism algebra over $\F_{5}$The isogeny class factors as 1.5.ae $\times$ 1.5.ac $\times$ 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 $\F_{5^{12}}$ is 1.244140625.abiuc 2 $\times$ 1.244140625.qkk 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 $\times$ 2.25.ac_av. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{5^{3}}$
The base change of $A$ to $\F_{5^{3}}$ is 1.125.ae $\times$ 1.125.w $\times$ 2.125.a_afm. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{5^{4}}$
The base change of $A$ to $\F_{5^{4}}$ is 1.625.o 2 $\times$ 2.625.abu_cfj. The endomorphism algebra for each factor is: - 1.625.o 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 2.625.abu_cfj : \(\Q(\sqrt{2}, \sqrt{-3})\).
- Endomorphism algebra over $\F_{5^{6}}$
The base change of $A$ to $\F_{5^{6}}$ is 1.15625.aja $\times$ 1.15625.afm 2 $\times$ 1.15625.ja. The endomorphism algebra for each factor is: - 1.15625.aja : \(\Q(\sqrt{-1}) \).
- 1.15625.afm 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$
- 1.15625.ja : \(\Q(\sqrt{-1}) \).
Base change
This is a primitive isogeny class.