Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 7 x + 25 x^{2} - 56 x^{3} + 64 x^{4}$ |
Frobenius angles: | $\pm0.113218980851$, $\pm0.403003401001$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.19097.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $27$ | $4131$ | $269568$ | $16528131$ | $1064375667$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $2$ | $66$ | $527$ | $4034$ | $32482$ | $262431$ | $2101906$ | $16791298$ | $134231855$ | $1073736546$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+a^2x^5+a^2x^4+ax^3+(a^2+a+1)x^2+ax+a^2+a+1$
- $y^2+(x^3+(a^2+a+1)x+a^2+a+1)y=x^6+(a^2+a)x^5+(a^2+a)x^4+a^2x^3+(a+1)x^2+a^2x+a+1$
- $y^2+(x^3+(a+1)x+a+1)y=x^6+ax^5+ax^4+(a^2+a)x^3+(a^2+1)x^2+(a^2+a)x+a^2+1$
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 4.0.19097.1. |
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.h_z | $2$ | 2.64.b_abf |