Invariants
Base field: | $\F_{5}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 4 x + 5 x^{2} )( 1 - 7 x + 25 x^{2} - 63 x^{3} + 125 x^{4} - 175 x^{5} + 125 x^{6} )$ |
$1 - 11 x + 58 x^{2} - 198 x^{3} + 502 x^{4} - 990 x^{5} + 1450 x^{6} - 1375 x^{7} + 625 x^{8}$ | |
Frobenius angles: | $\pm0.0923731703714$, $\pm0.147583617650$, $\pm0.243942915084$, $\pm0.536165446792$ |
Angle rank: | $4$ (numerical) |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $4$ |
Slopes: | $[0, 0, 0, 0, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $62$ | $323020$ | $227332238$ | $152331063680$ | $102004742170342$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-5$ | $21$ | $115$ | $625$ | $3335$ | $16125$ | $78192$ | $390529$ | $1957075$ | $9773621$ |
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 $\times$ 3.5.ah_z_acl 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.