Invariants
Base field: | $\F_{3}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )( 1 - x + 3 x^{2} )$ |
$1 - 6 x + 20 x^{2} - 42 x^{3} + 60 x^{4} - 54 x^{5} + 27 x^{6}$ | |
Frobenius angles: | $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.406785250661$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $0$ |
Isomorphism classes: | 2 |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $6$ | $1260$ | $38304$ | $655200$ | $13968966$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-2$ | $14$ | $46$ | $98$ | $238$ | $728$ | $2266$ | $6722$ | $19738$ | $58814$ |
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 $\times$ 1.3.ac $\times$ 1.3.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.abu $\times$ 1.729.ak $\times$ 1.729.cc. 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 $\times$ 1.9.c $\times$ 1.9.f. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 1.27.i $\times$ 1.27.k. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.