Invariants
Base field: | $\F_{3}$ |
Dimension: | $5$ |
L-polynomial: | $( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}( 1 - 3 x + 7 x^{2} - 9 x^{3} + 9 x^{4} )$ |
$1 - 11 x + 61 x^{2} - 221 x^{3} + 579 x^{4} - 1146 x^{5} + 1737 x^{6} - 1989 x^{7} + 1647 x^{8} - 891 x^{9} + 243 x^{10}$ | |
Frobenius angles: | $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.227267020856$, $\pm0.304086723985$, $\pm0.464830336654$ |
Angle rank: | $3$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $10$ | $85260$ | $31430560$ | $5187218400$ | $1101908764000$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-7$ | $11$ | $47$ | $115$ | $308$ | $857$ | $2303$ | $6451$ | $19361$ | $58946$ |
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 2 $\times$ 1.3.ac $\times$ 2.3.ad_h 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.abu $\times$ 1.729.cc 2 $\times$ 2.729.cn_dov. 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 2 $\times$ 1.9.c $\times$ 2.9.f_n. The endomorphism algebra for each factor is: - 1.9.ad 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 1.9.c : \(\Q(\sqrt{-2}) \).
- 2.9.f_n : 4.0.1525.1.
- Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 1.27.k $\times$ 2.27.j_cv. The endomorphism algebra for each factor is: - 1.27.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 1.27.k : \(\Q(\sqrt{-2}) \).
- 2.27.j_cv : 4.0.1525.1.
Base change
This is a primitive isogeny class.