Invariants
| Base field: | $\F_{3^{2}}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 5 x + 9 x^{2} )( 1 - 7 x + 25 x^{2} - 63 x^{3} + 81 x^{4} )$ |
| $1 - 12 x + 69 x^{2} - 251 x^{3} + 621 x^{4} - 972 x^{5} + 729 x^{6}$ | |
| Frobenius angles: | $\pm0.0842035494981$, $\pm0.186429498677$, $\pm0.435433986784$ |
| Angle rank: | $3$ (numerical) |
| Isomorphism classes: | 10 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $185$ | $491175$ | $389699540$ | $279117561375$ | $205690848972800$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-2$ | $76$ | $733$ | $6484$ | $58993$ | $533329$ | $4791022$ | $43054564$ | $387390037$ | $3486728551$ |
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.ah_z 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.ac_ab_ab | $2$ | (not in LMFDB) |
| 3.9.c_ab_b | $2$ | (not in LMFDB) |
| 3.9.m_cr_jr | $2$ | (not in LMFDB) |