Invariants
Base field: | $\F_{5}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 4 x + 5 x^{2} )( 1 - 4 x + 9 x^{2} - 20 x^{3} + 25 x^{4} )$ |
$1 - 8 x + 30 x^{2} - 76 x^{3} + 150 x^{4} - 200 x^{5} + 125 x^{6}$ | |
Frobenius angles: | $\pm0.103885594917$, $\pm0.147583617650$, $\pm0.516810247272$ |
Angle rank: | $3$ (numerical) |
Jacobians: | $0$ |
Isomorphism classes: | 2 |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $22$ | $12980$ | $1696288$ | $226371200$ | $31650524422$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-2$ | $22$ | $106$ | $578$ | $3238$ | $16192$ | $78846$ | $391618$ | $1958578$ | $9777702$ |
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 $\F_{5}$.
Endomorphism algebra over $\F_{5}$The isogeny class factors as 1.5.ae $\times$ 2.5.ae_j 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.