Invariants
Base field: | $\F_{3}$ |
Dimension: | $5$ |
L-polynomial: | $( 1 - 3 x + 3 x^{2} )( 1 - 7 x + 26 x^{2} - 67 x^{3} + 131 x^{4} - 201 x^{5} + 234 x^{6} - 189 x^{7} + 81 x^{8} )$ |
$1 - 10 x + 50 x^{2} - 166 x^{3} + 410 x^{4} - 795 x^{5} + 1230 x^{6} - 1494 x^{7} + 1350 x^{8} - 810 x^{9} + 243 x^{10}$ | |
Frobenius angles: | $\pm0.0365491909691$, $\pm0.166666666667$, $\pm0.234353293111$, $\pm0.355928212636$, $\pm0.548250857189$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $9$ | $59031$ | $15595524$ | $3463289739$ | $933273474489$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-6$ | $10$ | $30$ | $82$ | $269$ | $742$ | $2080$ | $6482$ | $19776$ | $58315$ |
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 $\times$ 4.3.ah_ba_acp_fb 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.cc $\times$ 4.729.abq_boh_acfrw_djcin. 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 $\times$ 4.9.d_a_ax_adl. 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 $\times$ 4.27.c_at_ak_ml. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.