Invariants
Base field: | $\F_{3}$ |
Dimension: | $5$ |
L-polynomial: | $( 1 - 3 x + 3 x^{2} )^{3}( 1 - x - x^{2} - 3 x^{3} + 9 x^{4} )$ |
$1 - 10 x + 44 x^{2} - 111 x^{3} + 189 x^{4} - 297 x^{5} + 567 x^{6} - 999 x^{7} + 1188 x^{8} - 810 x^{9} + 243 x^{10}$ | |
Frobenius angles: | $\pm0.126866938441$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.718153680921$ |
Angle rank: | $2$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $5$ | $22295$ | $9988160$ | $6122764375$ | $1202111664400$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-6$ | $-2$ | $15$ | $126$ | $329$ | $901$ | $2612$ | $6854$ | $19905$ | $59053$ |
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 3 $\times$ 2.3.ab_ab 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 3 $\times$ 2.729.j_jt. The endomorphism algebra for each factor is:
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- Endomorphism algebra over $\F_{3^{2}}$
The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad 3 $\times$ 2.9.ad_n. The endomorphism algebra for each factor is: - 1.9.ad 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
- 2.9.ad_n : 4.0.10933.1.
- Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 3 $\times$ 2.27.an_dl. The endomorphism algebra for each factor is: - 1.27.a 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
- 2.27.an_dl : 4.0.10933.1.
Base change
This is a primitive isogeny class.