Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 3 x + 7 x^{2} - 24 x^{3} + 64 x^{4}$ |
Frobenius angles: | $\pm0.171649807387$, $\pm0.606294418218$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.6025.1 |
Galois group: | $D_{4}$ |
Jacobians: | $15$ |
Isomorphism classes: | 21 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $45$ | $4455$ | $244620$ | $16951275$ | $1096437375$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $70$ | $477$ | $4138$ | $33456$ | $262735$ | $2097402$ | $16787698$ | $134217621$ | $1073636350$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 15 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+(a+1)x+a+1)y=(a+1)x^3+(a^2+1)x+a^2+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^2+1)x^3+(a^2+a+1)x+a^2+a+1$
- $y^2+(x^3+(a^2+a+1)x+a^2+a+1)y=(a^2+a+1)x^3+(a+1)x+a+1$
- $y^2+(x^3+(a^2+a+1)x+a^2+a+1)y=ax^5+ax^4+(a^2+a+1)x^3+a^2+1$
- $y^2+(x^3+(a^2+a+1)x+a^2+a+1)y=x^6+ax^5+ax^4+(a^2+a)x^3+(a+1)x^2+(a^2+a+1)x+1$
- $y^2+(x^3+(a+1)x+a+1)y=x^6+ax^3+(a^2+1)x^2+(a^2+a)x+a+1$
- $y^2+(x^3+(a+1)x+a+1)y=a^2x^5+a^2x^4+(a+1)x^3+a^2+a+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+a^2x^3+(a^2+a+1)x^2+ax+a^2+1$
- $y^2+(x^3+(a+1)x+a+1)y=x^6+a^2x^5+a^2x^4+ax^3+(a^2+1)x^2+(a+1)x+1$
- $y^2+(x^3+(a+1)x+a+1)y=x^6+(a+1)x^5+(a+1)x^4+(a^2+1)x^3+(a^2+1)x^2+(a^2+1)x+a^2+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^2+a)x^5+(a^2+a)x^4+(a^2+1)x^3+a+1$
- $y^2+(x^3+(a^2+a+1)x+a^2+a+1)y=x^6+(a^2+a)x^3+(a+1)x^2+a^2x+a^2+a+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+(a^2+a)x^5+(a^2+a)x^4+a^2x^3+(a^2+a+1)x^2+(a^2+1)x+1$
- $y^2+(x^3+(a^2+a+1)x+a^2+a+1)y=x^6+(a^2+a+1)x^5+(a^2+a+1)x^4+(a+1)x^3+(a+1)x^2+(a+1)x+a+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+(a^2+1)x^5+(a^2+1)x^4+(a^2+a+1)x^3+(a^2+a+1)x^2+(a^2+a+1)x+a^2+a+1$
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 4.0.6025.1. |
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.d_h | $2$ | 2.64.f_bh |