Invariants
Base field: | $\F_{5^{4}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 25 x )^{2}( 1 - 49 x + 625 x^{2} )$ |
$1 - 99 x + 3700 x^{2} - 61875 x^{3} + 390625 x^{4}$ | |
Frobenius angles: | $0$, $0$, $\pm0.0637685608585$ |
Angle rank: | $1$ (numerical) |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $332352$ | $151652217600$ | $59590724194052352$ | $23282862217526698675200$ | $9094944151318564946239474752$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $527$ | $388225$ | $244083602$ | $152586565825$ | $95367401584127$ | $59604644111091550$ | $37252902970365516527$ | $23283064365091515201025$ | $14551915228360978209130802$ | $9094947017729171598283878625$ |
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 $\F_{5^{4}}$.
Endomorphism algebra over $\F_{5^{4}}$The isogeny class factors as 1.625.aby $\times$ 1.625.abx 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.