Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 3 x )^{2}( 1 - 5 x + 9 x^{2} )^{2}$ |
$1 - 16 x + 112 x^{2} - 438 x^{3} + 1008 x^{4} - 1296 x^{5} + 729 x^{6}$ | |
Frobenius angles: | $0$, $0$, $\pm0.186429498677$, $\pm0.186429498677$ |
Angle rank: | $1$ (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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $100$ | $360000$ | $370177600$ | $285156000000$ | $207505465502500$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-6$ | $50$ | $696$ | $6626$ | $59514$ | $532700$ | $4783626$ | $43034306$ | $387339384$ | $3486451250$ |
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 2 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.