Invariants
Base field: | $\F_{3}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 3 x + 3 x^{2} )^{2}( 1 - x + x^{2} - 3 x^{3} + 9 x^{4} )$ |
$1 - 7 x + 22 x^{2} - 42 x^{3} + 69 x^{4} - 126 x^{5} + 198 x^{6} - 189 x^{7} + 81 x^{8}$ | |
Frobenius angles: | $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.201748855633$, $\pm0.672988571819$ |
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: | $[0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $7$ | $5145$ | $433552$ | $73907925$ | $5190222352$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-3$ | $5$ | $21$ | $125$ | $342$ | $827$ | $2433$ | $6725$ | $19173$ | $58400$ |
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 $\times$ 2.3.ab_b 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 2 $\times$ 2.729.al_abfv. 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 2 $\times$ 2.9.b_n. The endomorphism algebra for each factor is: - 1.9.ad 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 2.9.b_n : 4.0.16317.1.
- Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 2.27.ah_t. The endomorphism algebra for each factor is: - 1.27.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 2.27.ah_t : 4.0.16317.1.
Base change
This is a primitive isogeny class.