Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 5 x + 21 x^{2} - 40 x^{3} + 64 x^{4}$ |
Frobenius angles: | $\pm0.279106592915$, $\pm0.421442037109$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.14225.1 |
Galois group: | $D_{4}$ |
Jacobians: | $3$ |
Isomorphism classes: | 3 |
This isogeny class is simple and geometrically 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $41$ | $5371$ | $300284$ | $16977731$ | $1066394461$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $82$ | $583$ | $4146$ | $32544$ | $261703$ | $2097148$ | $16774338$ | $134200159$ | $1073742522$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+(a^2+a+1)x+a^2+a+1)y=x^6+x^5+x^4+(a^2+1)x^3+(a+1)x^2+ax+a^2+a$
- $y^2+(x^3+(a+1)x+a+1)y=x^6+x^5+x^4+(a^2+a+1)x^3+(a^2+1)x^2+a^2x+a$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+x^5+x^4+(a+1)x^3+(a^2+a+1)x^2+(a^2+a)x+a^2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{3}}$.
Endomorphism algebra over $\F_{2^{3}}$The endomorphism algebra of this simple isogeny class is 4.0.14225.1. |
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.f_v | $2$ | 2.64.r_gn |