Invariants
Base field: | $\F_{3}$ |
Dimension: | $5$ |
L-polynomial: | $( 1 - 3 x + 3 x^{2} )( 1 - 2 x + 3 x^{2} )( 1 - 5 x + 13 x^{2} - 25 x^{3} + 39 x^{4} - 45 x^{5} + 27 x^{6} )$ |
$1 - 10 x + 50 x^{2} - 165 x^{3} + 404 x^{4} - 780 x^{5} + 1212 x^{6} - 1485 x^{7} + 1350 x^{8} - 810 x^{9} + 243 x^{10}$ | |
Frobenius angles: | $\pm0.0714477711956$, $\pm0.166666666667$, $\pm0.272071776080$, $\pm0.304086723985$, $\pm0.560185743604$ |
Angle rank: | $4$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $4$ |
Slopes: | $[0, 0, 0, 0, 1/2, 1/2, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $10$ | $65100$ | $17401720$ | $4231500000$ | $1047034665050$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-6$ | $10$ | $33$ | $98$ | $294$ | $727$ | $2003$ | $6498$ | $20229$ | $59750$ |
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 $\F_{3^{6}}$.
Endomorphism algebra over $\F_{3}$The isogeny class factors as 1.3.ad $\times$ 1.3.ac $\times$ 3.3.af_n_az and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
The base change of $A$ to $\F_{3^{6}}$ is 1.729.abu $\times$ 1.729.cc $\times$ 3.729.al_cmn_aoxb. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{3^{2}}$
The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad $\times$ 1.9.c $\times$ 3.9.b_ad_ah. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is 1.27.a $\times$ 1.27.k $\times$ 3.27.af_h_ev. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.