Invariants
Base field: | $\F_{3}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 3 x + 3 x^{2} )( 1 - 4 x + 6 x^{2} - 7 x^{3} + 18 x^{4} - 36 x^{5} + 27 x^{6} )$ |
$1 - 7 x + 21 x^{2} - 37 x^{3} + 57 x^{4} - 111 x^{5} + 189 x^{6} - 189 x^{7} + 81 x^{8}$ | |
Frobenius angles: | $\pm0.0452398905210$, $\pm0.166666666667$, $\pm0.239335307006$, $\pm0.691360448188$ |
Angle rank: | $3$ (numerical) |
Isomorphism classes: | 1 |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1/2, 1/2, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $5$ | $3465$ | $321020$ | $57882825$ | $3993666025$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-3$ | $3$ | $15$ | $107$ | $277$ | $711$ | $2251$ | $6403$ | $19140$ | $59043$ |
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 $\F_{3^{6}}$.
Endomorphism algebra over $\F_{3}$The isogeny class factors as 1.3.ad $\times$ 3.3.ae_g_ah 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 $\F_{3^{6}}$ is 1.729.cc $\times$ 3.729.acv_eli_afacr. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{3^{2}}$
The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad $\times$ 3.9.ae_q_acp. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 3.27.an_bw_acv. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.