Invariants
Base field: | $\F_{5}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 4 x + 5 x^{2} )( 1 - 7 x + 27 x^{2} - 73 x^{3} + 135 x^{4} - 175 x^{5} + 125 x^{6} )$ |
$1 - 11 x + 60 x^{2} - 216 x^{3} + 562 x^{4} - 1080 x^{5} + 1500 x^{6} - 1375 x^{7} + 625 x^{8}$ | |
Frobenius angles: | $\pm0.0229621162481$, $\pm0.147583617650$, $\pm0.333082169302$, $\pm0.478604549684$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $66$ | $358380$ | $245992626$ | $139189057920$ | $91763628038346$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-5$ | $25$ | $127$ | $569$ | $3005$ | $15613$ | $78164$ | $390177$ | $1953541$ | $9769465$ |
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_bb_acv 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.