Invariants
Base field: | $\F_{3}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 + 2 x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{3}$ |
$1 - 7 x + 21 x^{2} - 36 x^{3} + 54 x^{4} - 108 x^{5} + 189 x^{6} - 189 x^{7} + 81 x^{8}$ | |
Frobenius angles: | $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.695913276015$ |
Angle rank: | $1$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $6$ | $4116$ | $395136$ | $72342816$ | $4896017706$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-3$ | $3$ | $18$ | $123$ | $327$ | $846$ | $2517$ | $6771$ | $19494$ | $58803$ |
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 $\times$ 1.3.c 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 3 . 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 $\times$ 1.9.c. The endomorphism algebra for each factor is: - 1.9.ad 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
- 1.9.c : \(\Q(\sqrt{-2}) \).
- Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is 1.27.ak $\times$ 1.27.a 3 . The endomorphism algebra for each factor is: - 1.27.ak : \(\Q(\sqrt{-2}) \).
- 1.27.a 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
Base change
This is a primitive isogeny class.