Invariants
Base field: | $\F_{3}$ |
Dimension: | $5$ |
L-polynomial: | $( 1 - 2 x + 3 x^{2} )( 1 - x + 3 x^{2} )( 1 - 3 x + 3 x^{2} )^{3}$ |
$1 - 12 x + 71 x^{2} - 270 x^{3} + 729 x^{4} - 1458 x^{5} + 2187 x^{6} - 2430 x^{7} + 1917 x^{8} - 972 x^{9} + 243 x^{10}$ | |
Frobenius angles: | $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.166666666667$, $\pm0.304086723985$, $\pm0.406785250661$ |
Angle rank: | $2$ (numerical) |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1/2, 1/2, 1/2, 1/2, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $6$ | $61740$ | $30030336$ | $5425711200$ | $1025894832006$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-8$ | $8$ | $46$ | $116$ | $292$ | $836$ | $2428$ | $6884$ | $19738$ | $58328$ |
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 3 $\times$ 1.3.ac $\times$ 1.3.ab 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.ak $\times$ 1.729.cc 3 . The endomorphism algebra for each factor is:
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- Endomorphism algebra over $\F_{3^{2}}$
The base change of $A$ to $\F_{3^{2}}$ is 1.9.ad 3 $\times$ 1.9.c $\times$ 1.9.f. The endomorphism algebra for each factor is: - 1.9.ad 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
- 1.9.c : \(\Q(\sqrt{-2}) \).
- 1.9.f : \(\Q(\sqrt{-11}) \).
- Endomorphism algebra over $\F_{3^{3}}$
The base change of $A$ to $\F_{3^{3}}$ is 1.27.a 3 $\times$ 1.27.i $\times$ 1.27.k. The endomorphism algebra for each factor is: - 1.27.a 3 : $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$
- 1.27.i : \(\Q(\sqrt{-11}) \).
- 1.27.k : \(\Q(\sqrt{-2}) \).
Base change
This is a primitive isogeny class.