Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 3 x + 5 x^{2} )( 1 - 4 x + 11 x^{2} - 20 x^{3} + 25 x^{4} )$ |
| $1 - 7 x + 28 x^{2} - 73 x^{3} + 140 x^{4} - 175 x^{5} + 125 x^{6}$ | |
| Frobenius angles: | $\pm0.185749715683$, $\pm0.265942140215$, $\pm0.480916950984$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $0$ |
| Isomorphism classes: | 6 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $39$ | $21411$ | $2433600$ | $257467275$ | $31848630099$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-1$ | $33$ | $152$ | $661$ | $3259$ | $16020$ | $78175$ | $388421$ | $1949336$ | $9768153$ |
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 $\F_{5^{3}}$.
Endomorphism algebra over $\F_{5}$| The isogeny class factors as 1.5.ad $\times$ 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 $\F_{5^{3}}$ is 1.125.e 2 $\times$ 1.125.s. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.