Invariants
Base field: | $\F_{3}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - x + 3 x^{2} )( 1 + 2 x + 3 x^{2} )$ |
$1 + x + 4 x^{2} + 3 x^{3} + 9 x^{4}$ | |
Frobenius angles: | $\pm0.406785250661$, $\pm0.695913276015$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $1$ |
Isomorphism classes: | 5 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $18$ | $180$ | $648$ | $7200$ | $52398$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $17$ | $26$ | $89$ | $215$ | $674$ | $2357$ | $6641$ | $19358$ | $59057$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+x^4+2x^3+x^2+2x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3}$.
Endomorphism algebra over $\F_{3}$The isogeny class factors as 1.3.ab $\times$ 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: |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.3.ad_i | $2$ | 2.9.h_bc |
2.3.ab_e | $2$ | 2.9.h_bc |
2.3.d_i | $2$ | 2.9.h_bc |