Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 2 x + 3 x^{2} )( 1 - 3 x + 7 x^{2} - 9 x^{3} + 9 x^{4} )$ |
| $1 - 5 x + 16 x^{2} - 32 x^{3} + 48 x^{4} - 45 x^{5} + 27 x^{6}$ | |
| Frobenius angles: | $\pm0.227267020856$, $\pm0.304086723985$, $\pm0.464830336654$ |
| Angle rank: | $3$ (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: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $10$ | $1740$ | $40090$ | $626400$ | $15004000$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-1$ | $17$ | $47$ | $97$ | $254$ | $749$ | $2141$ | $6289$ | $19361$ | $59432$ |
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 $\times$ 2.3.ad_h 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 |
|---|---|---|
| 3.3.ab_e_ae | $2$ | 3.9.h_bg_em |
| 3.3.b_e_e | $2$ | 3.9.h_bg_em |
| 3.3.f_q_bg | $2$ | 3.9.h_bg_em |