Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 4 x + 5 x^{2} )( 1 - x + 5 x^{2} )$ |
$1 - 5 x + 14 x^{2} - 25 x^{3} + 25 x^{4}$ | |
Frobenius angles: | $\pm0.147583617650$, $\pm0.428216853436$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $1$ |
Isomorphism classes: | 3 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $10$ | $700$ | $17080$ | $380800$ | $9686050$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $1$ | $29$ | $136$ | $609$ | $3101$ | $15914$ | $79241$ | $391969$ | $1952056$ | $9762149$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+x^4+x+2$
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 $\times$ 1.5.ab 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.