Invariants
Base field: | $\F_{5^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 9 x + 25 x^{2} )( 1 - 6 x + 25 x^{2} )$ |
$1 - 15 x + 104 x^{2} - 375 x^{3} + 625 x^{4}$ | |
Frobenius angles: | $\pm0.143566293129$, $\pm0.295167235301$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $7$ |
Isomorphism classes: | 18 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $340$ | $380800$ | $246971920$ | $153113587200$ | $95410665939700$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $11$ | $609$ | $15806$ | $391969$ | $9770051$ | $244145454$ | $6103518971$ | $152588258689$ | $3814701483566$ | $95367454868049$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 7 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+4)x^6+(3a+1)x^5+(4a+3)x^4+(a+2)x+a+2$
- $y^2=ax^6+3x^5+(a+2)x^4+4x^3+ax^2+(4a+1)x+2a+3$
- $y^2=(2a+4)x^6+(2a+3)x^5+(4a+3)x^4+x^3+(4a+2)x^2+(a+1)x+a+2$
- $y^2=(3a+4)x^6+(3a+4)x^5+ax^4+(a+4)x^3+4ax^2+(3a+4)x+4a+3$
- $y^2=(4a+1)x^6+(3a+2)x^5+(a+4)x^4+(4a+2)x^3+x^2+(3a+2)x+a+2$
- $y^2=(2a+4)x^6+(3a+3)x^5+(2a+3)x^4+2x^3+(4a+1)x^2+(4a+3)x+3a$
- $y^2=(a+4)x^6+(3a+4)x^5+3x^4+3ax^3+3ax^2+(4a+3)x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{2}}$.
Endomorphism algebra over $\F_{5^{2}}$The isogeny class factors as 1.25.aj $\times$ 1.25.ag 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.