Invariants
Base field: | $\F_{3^{4}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 9 x )^{2}( 1 - 17 x + 81 x^{2} )$ |
$1 - 35 x + 468 x^{2} - 2835 x^{3} + 6561 x^{4}$ | |
Frobenius angles: | $0$, $0$, $\pm0.106600758076$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $0$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $4160$ | $41184000$ | $281241309440$ | $1852326013824000$ | $12157296301288424000$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $47$ | $6273$ | $529202$ | $43030593$ | $3486678527$ | $282428924958$ | $22876789572047$ | $1853020179809793$ | $150094635290706962$ | $12157665458907527073$ |
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 $\F_{3^{4}}$.
Endomorphism algebra over $\F_{3^{4}}$The isogeny class factors as 1.81.as $\times$ 1.81.ar 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.