Invariants
| Base field: | $\F_{3^{3}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 8 x + 27 x^{2} )( 1 - 7 x + 27 x^{2} )$ |
| $1 - 15 x + 110 x^{2} - 405 x^{3} + 729 x^{4}$ | |
| Frobenius angles: | $\pm0.220355751984$, $\pm0.264757707515$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $0$ |
| Isomorphism classes: | 12 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $420$ | $529200$ | $394576560$ | $283915800000$ | $206051538485100$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $13$ | $725$ | $20044$ | $534233$ | $14360083$ | $387430550$ | $10460142769$ | $282427764593$ | $7625589982228$ | $205891121901125$ |
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 $\F_{3^{3}}$.
Endomorphism algebra over $\F_{3^{3}}$| The isogeny class factors as 1.27.ai $\times$ 1.27.ah 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 |
|---|---|---|
| 2.27.ab_ac | $2$ | 2.729.af_cce |
| 2.27.b_ac | $2$ | 2.729.af_cce |
| 2.27.p_eg | $2$ | 2.729.af_cce |