Invariants
Base field: | $\F_{3}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 2 x + 3 x^{2} )( 1 - 2 x + 5 x^{2} - 6 x^{3} + 9 x^{4} )$ |
$1 - 4 x + 12 x^{2} - 22 x^{3} + 36 x^{4} - 36 x^{5} + 27 x^{6}$ | |
Frobenius angles: | $\pm0.254551732336$, $\pm0.304086723985$, $\pm0.538152604671$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $14$ | $1932$ | $32984$ | $633696$ | $16738414$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $18$ | $42$ | $98$ | $280$ | $720$ | $1960$ | $6274$ | $19950$ | $59778$ |
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.ac_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.a_e_ac | $2$ | 3.9.i_bo_fq |
3.3.a_e_c | $2$ | 3.9.i_bo_fq |
3.3.e_m_w | $2$ | 3.9.i_bo_fq |