Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 3 x )^{2}( 1 - 5 x + 9 x^{2} )( 1 - x + 9 x^{2} )$ |
$1 - 12 x + 68 x^{2} - 246 x^{3} + 612 x^{4} - 972 x^{5} + 729 x^{6}$ | |
Frobenius angles: | $0$, $0$, $\pm0.186429498677$, $\pm0.446699620962$ |
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: | $[0, 0, 1/2, 1/2, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $180$ | $475200$ | $378181440$ | $274903200000$ | $204591151026900$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-2$ | $74$ | $712$ | $6386$ | $58678$ | $532124$ | $4785142$ | $43030946$ | $387320968$ | $3486546554$ |
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 $\F_{3^{2}}$.
Endomorphism algebra over $\F_{3^{2}}$The isogeny class factors as 1.9.ag $\times$ 1.9.af $\times$ 1.9.ab 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.