Invariants
Base field: | $\F_{3}$ |
Dimension: | $5$ |
L-polynomial: | $( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{2}( 1 - 3 x + 5 x^{2} - 9 x^{3} + 9 x^{4} )$ |
$1 - 11 x + 59 x^{2} - 205 x^{3} + 519 x^{4} - 1014 x^{5} + 1557 x^{6} - 1845 x^{7} + 1593 x^{8} - 891 x^{9} + 243 x^{10}$ | |
Frobenius angles: | $\pm0.0975263560046$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.527857038681$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $6$ | $47628$ | $16355808$ | $3927976416$ | $1109870943456$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-7$ | $7$ | $29$ | $91$ | $308$ | $853$ | $2261$ | $6691$ | $20333$ | $59842$ |
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_f 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.cj_ddt. 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.b_al. 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.b_al : 4.0.2197.1.
- 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.aj_ct. 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.aj_ct : 4.0.2197.1.
Base change
This is a primitive isogeny class.