Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 3 x + 8 x^{2} )( 1 - x + 8 x^{2} )$ |
$1 - 4 x + 19 x^{2} - 32 x^{3} + 64 x^{4}$ | |
Frobenius angles: | $\pm0.322067999368$, $\pm0.443432958871$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $9$ |
Isomorphism classes: | 36 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $48$ | $5760$ | $299088$ | $16704000$ | $1060603248$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $87$ | $581$ | $4079$ | $32365$ | $261639$ | $2097877$ | $16777951$ | $134216573$ | $1073778807$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x+1)y=ax^5+ax^3+x^2+(a+1)x+1$
- $y^2+(x^2+x+1)y=a^2x^5+a^2x^3+(a+1)x^2+(a^2+a+1)x+a+1$
- $y^2+(x^2+x+1)y=a^2x^5+a^2x^3+x^2+(a^2+1)x+1$
- $y^2+(x^2+x+1)y=(a^2+a)x^5+(a^2+a)x^3+(a^2+1)x^2+(a+1)x+a^2+1$
- $y^2+(x^2+x+1)y=(a^2+a)x^5+x^4+(a^2+a)x^3+ax^2+(a^2+1)x+a$
- $y^2+(x^2+x+1)y=(a^2+a)x^5+(a^2+a)x^3+x^2+(a^2+a+1)x+1$
- $y^2+(x^2+x+1)y=ax^5+ax^3+(a^2+a+1)x^2+(a^2+1)x+a^2+a+1$
- $y^2+(x^2+x+1)y=a^2x^5+x^4+a^2x^3+(a^2+a)x^2+(a+1)x+a^2+a$
- $y^2+(x^2+x+1)y=ax^5+x^4+ax^3+a^2x^2+(a^2+a+1)x+a^2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{3}}$.
Endomorphism algebra over $\F_{2^{3}}$The isogeny class factors as 1.8.ad $\times$ 1.8.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.
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.8.ac_n | $2$ | 2.64.w_iz |
2.8.c_n | $2$ | 2.64.w_iz |
2.8.e_t | $2$ | 2.64.w_iz |