Invariants
Base field: | $\F_{5^{4}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 48 x + 625 x^{2} )( 1 - 46 x + 625 x^{2} )$ |
$1 - 94 x + 3458 x^{2} - 58750 x^{3} + 390625 x^{4}$ | |
Frobenius angles: | $\pm0.0903344706017$, $\pm0.128188433698$ |
Angle rank: | $2$ (numerical) |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $335240$ | $151839582720$ | $59596911605204360$ | $23283018837727720243200$ | $9094947534077905730831256200$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $532$ | $388706$ | $244108948$ | $152587592254$ | $95367437054932$ | $59604645205023266$ | $37252903001126802388$ | $23283064365887655467134$ | $14551915228379959368778132$ | $9094947017729585094940269026$ |
Jacobians and polarizations
This isogeny class contains a Jacobian, and hence is principally polarizable.
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.abw $\times$ 1.625.abu 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.