Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 4 x + 5 x^{2} )( 1 - 3 x + 8 x^{2} - 15 x^{3} + 25 x^{4} )$ |
| $1 - 7 x + 25 x^{2} - 62 x^{3} + 125 x^{4} - 175 x^{5} + 125 x^{6}$ | |
| Frobenius angles: | $\pm0.147583617650$, $\pm0.206741677780$, $\pm0.540075011113$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $0$ |
| Isomorphism classes: | 12 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $32$ | $16640$ | $1928576$ | $249200640$ | $33084652192$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-1$ | $27$ | $122$ | $639$ | $3379$ | $16224$ | $78343$ | $390431$ | $1954562$ | $9761907$ |
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.ad_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 $\F_{5^{6}}$ is 1.15625.ha 2 $\times$ 1.15625.ja. 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 $\times$ 2.25.h_y. 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 $\times$ 2.125.a_ha. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.