Invariants
Base field: | $\F_{3}$ |
Dimension: | $5$ |
L-polynomial: | $( 1 - 3 x + 3 x^{2} )^{2}( 1 - 5 x + 15 x^{2} - 31 x^{3} + 45 x^{4} - 45 x^{5} + 27 x^{6} )$ |
$1 - 11 x + 60 x^{2} - 214 x^{3} + 555 x^{4} - 1095 x^{5} + 1665 x^{6} - 1926 x^{7} + 1620 x^{8} - 891 x^{9} + 243 x^{10}$ | |
Frobenius angles: | $\pm0.113296540390$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.351823865540$, $\pm0.481790494592$ |
Angle rank: | $3$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $7$ | $57967$ | $20519632$ | $3693251471$ | $957574946377$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-7$ | $9$ | $35$ | $85$ | $273$ | $867$ | $2450$ | $6869$ | $20069$ | $59389$ |
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 2 $\times$ 3.3.af_p_abf 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 2 $\times$ 3.729.bd_apb_abvut. 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 2 $\times$ 3.9.f_f_ah. The endomorphism algebra for each factor is: - 1.9.ad 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 3.9.f_f_ah : 6.0.400967.1.
- Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 2 $\times$ 3.27.h_bn_lb. The endomorphism algebra for each factor is: - 1.27.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 3.27.h_bn_lb : 6.0.400967.1.
Base change
This is a primitive isogeny class.