Invariants
Base field: | $\F_{5}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 4 x + 5 x^{2} )( 1 - 4 x + 11 x^{2} - 20 x^{3} + 25 x^{4} )$ |
$1 - 8 x + 32 x^{2} - 84 x^{3} + 160 x^{4} - 200 x^{5} + 125 x^{6}$ | |
Frobenius angles: | $\pm0.147583617650$, $\pm0.185749715683$, $\pm0.480916950984$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $0$ |
Isomorphism classes: | 2 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $26$ | $15860$ | $2061800$ | $244117120$ | $32038741306$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-2$ | $26$ | $130$ | $626$ | $3278$ | $16328$ | $79238$ | $390626$ | $1951690$ | $9765626$ |
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.ae $\times$ 2.5.ae_l 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.ja 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-1}) \)$)$ |
- Endomorphism algebra over $\F_{5^{2}}$
The base change of $A$ to $\F_{5^{2}}$ is 1.25.ag $\times$ 2.25.g_l. 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.e 2 . The endomorphism algebra for each factor is: - 1.125.ae : \(\Q(\sqrt{-1}) \).
- 1.125.e 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
Base change
This is a primitive isogeny class.