Invariants
Base field: | $\F_{3^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 9 x + 27 x^{2} )( 1 - 4 x + 27 x^{2} )$ |
$1 - 13 x + 90 x^{2} - 351 x^{3} + 729 x^{4}$ | |
Frobenius angles: | $\pm0.166666666667$, $\pm0.374235869875$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $6$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $456$ | $539904$ | $392577696$ | $282825470976$ | $205885332491736$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $15$ | $741$ | $19944$ | $532185$ | $14348505$ | $387431622$ | $10460605443$ | $282431130609$ | $7625599708248$ | $205891097946261$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a^2+1)x^6+(a^2+a)x^4+2ax^3+2a^2x+2a^2+2a+1$
- $y^2=(a^2+a+2)x^6+x^4+ax^3+(2a^2+a+2)x+2a^2+a$
- $y^2=(a^2+2)x^6+(a^2+a+1)x^4+(2a^2+2a)x^3+(a^2+2a+1)x+2a^2+a$
- $y^2=(2a^2+a+2)x^6+x^4+(a^2+1)x^3+(a^2+a)x+2a^2+1$
- $y^2=(a^2+a+1)x^6+(2a^2+a+1)x^4+(2a+1)x^3+(a^2+a+2)x$
- $y^2=(2a^2+a+1)x^6+(2a+2)x^4+(a^2+2a+2)x^3+(2a^2+a)x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{18}}$.
Endomorphism algebra over $\F_{3^{3}}$The isogeny class factors as 1.27.aj $\times$ 1.27.ae 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 $\F_{3^{18}}$ is 1.387420489.abpty $\times$ 1.387420489.cggc. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{3^{6}}$
The base change of $A$ to $\F_{3^{6}}$ is 1.729.abb $\times$ 1.729.bm. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{3^{9}}$
The base change of $A$ to $\F_{3^{9}}$ is 1.19683.a $\times$ 1.19683.ka. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.