Invariants
Base field: | $\F_{5^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 9 x + 25 x^{2} )( 1 - 7 x + 25 x^{2} )$ |
$1 - 16 x + 113 x^{2} - 400 x^{3} + 625 x^{4}$ | |
Frobenius angles: | $\pm0.143566293129$, $\pm0.253183311107$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $6$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $323$ | $373065$ | $246162176$ | $153189815625$ | $95446998996563$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $596$ | $15754$ | $392164$ | $9773770$ | $244167086$ | $6103553914$ | $152587809604$ | $3814697136730$ | $95367437511476$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=4x^6+(4a+4)x^5+3ax^4+3x^3+3ax^2+(4a+4)x+4$
- $y^2=2ax^6+(3a+1)x^5+(3a+1)x^4+3x^3+(3a+1)x^2+(3a+1)x+2a$
- $y^2=4ax^6+4ax^5+(a+4)x^4+(4a+3)x^3+(a+4)x^2+4ax+4a$
- $y^2=(2a+3)x^6+(4a+3)x^5+(2a+3)x^4+(2a+1)x^3+(2a+3)x^2+(4a+3)x+2a+3$
- $y^2=(a+4)x^6+(a+3)x^5+3x^4+(2a+3)x^3+3x^2+(a+3)x+a+4$
- $y^2=(2a+3)x^6+2ax^5+(3a+3)x^4+3ax^3+(3a+3)x^2+2ax+2a+3$
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.ah 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.
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.25.ac_an | $2$ | 2.625.abe_bux |
2.25.c_an | $2$ | 2.625.abe_bux |
2.25.q_ej | $2$ | 2.625.abe_bux |