Invariants
Base field: | $\F_{3}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - x + 3 x^{2} )( 1 - 3 x + 5 x^{2} - 9 x^{3} + 9 x^{4} )$ |
$1 - 4 x + 11 x^{2} - 23 x^{3} + 33 x^{4} - 36 x^{5} + 27 x^{6}$ | |
Frobenius angles: | $\pm0.0975263560046$, $\pm0.406785250661$, $\pm0.527857038681$ |
Angle rank: | $3$ (numerical) |
Jacobians: | $0$ |
Isomorphism classes: | 5 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $9$ | $1215$ | $19764$ | $370575$ | $13301424$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $16$ | $27$ | $52$ | $225$ | $781$ | $2268$ | $6676$ | $20007$ | $59371$ |
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.ab $\times$ 2.3.ad_f 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.ac_f_an | $2$ | 3.9.g_d_abl |
3.3.c_f_n | $2$ | 3.9.g_d_abl |
3.3.e_l_x | $2$ | 3.9.g_d_abl |