Invariants
Base field: | $\F_{5^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 9 x + 25 x^{2} )( 1 - 5 x + 25 x^{2} )$ |
$1 - 14 x + 95 x^{2} - 350 x^{3} + 625 x^{4}$ | |
Frobenius angles: | $\pm0.143566293129$, $\pm0.333333333333$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $4$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $357$ | $387345$ | $247221072$ | $152945884665$ | $95375505232077$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $12$ | $620$ | $15822$ | $391540$ | $9766452$ | $244137710$ | $6103593732$ | $152588978980$ | $3814703545662$ | $95367445327100$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(4a+2)x^6+(3a+3)x^5+(3a+4)x^4+(4a+3)x^3+(3a+4)x^2+(3a+3)x+4a+2$
- $y^2=(a+4)x^6+x^5+x^4+2ax^3+(3a+3)x^2+(2a+2)x+a$
- $y^2=3x^6+(2a+2)x^5+x^4+(3a+2)x^3+x^2+(2a+2)x+3$
- $y^2=3ax^6+2x^5+(2a+1)x^4+(3a+3)x^3+(2a+1)x^2+3ax+2a$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{6}}$.
Endomorphism algebra over $\F_{5^{2}}$The isogeny class factors as 1.25.aj $\times$ 1.25.af and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
The base change of $A$ to $\F_{5^{6}}$ is 1.15625.acc $\times$ 1.15625.jq. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.