Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 2 x + 3 x^{2} )( 1 - x + 3 x^{2} )$ |
| $1 - 3 x + 8 x^{2} - 9 x^{3} + 9 x^{4}$ | |
| Frobenius angles: | $\pm0.304086723985$, $\pm0.406785250661$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $0$ |
| Isomorphism classes: | 1 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $2$ |
| Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6$ | $180$ | $1368$ | $7200$ | $51546$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $1$ | $17$ | $46$ | $89$ | $211$ | $674$ | $2185$ | $6641$ | $19738$ | $59057$ |
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 $\F_{3}$.
Endomorphism algebra over $\F_{3}$| The isogeny class factors as 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: |
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.ab_e | $2$ | 2.9.h_bc |
| 2.3.b_e | $2$ | 2.9.h_bc |
| 2.3.d_i | $2$ | 2.9.h_bc |