Invariants
| Base field: | $\F_{3^{2}}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 5 x + 9 x^{2} )( 1 - 8 x + 31 x^{2} - 72 x^{3} + 81 x^{4} )$ |
| $1 - 13 x + 80 x^{2} - 299 x^{3} + 720 x^{4} - 1053 x^{5} + 729 x^{6}$ | |
| Frobenius angles: | $\pm0.0954872438962$, $\pm0.186429498677$, $\pm0.376614839446$ |
| Angle rank: | $3$ (numerical) |
| Isomorphism classes: | 8 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $165$ | $477675$ | $401757840$ | $284710063275$ | $205843504466325$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-3$ | $73$ | $756$ | $6613$ | $59037$ | $532132$ | $4789509$ | $43069861$ | $387448596$ | $3486706153$ |
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 $\F_{3^{2}}$.
Endomorphism algebra over $\F_{3^{2}}$| The isogeny class factors as 1.9.af $\times$ 2.9.ai_bf 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 |
|---|---|---|
| 3.9.ad_a_l | $2$ | (not in LMFDB) |
| 3.9.d_a_al | $2$ | (not in LMFDB) |
| 3.9.n_dc_ln | $2$ | (not in LMFDB) |