Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 2 x - 3 x^{2} - 16 x^{3} + 64 x^{4}$ |
Frobenius angles: | $\pm0.0815683182843$, $\pm0.710356941346$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1025.1 |
Galois group: | $D_{4}$ |
Jacobians: | $10$ |
Isomorphism classes: | 20 |
This isogeny class is simple and geometrically simple, not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $44$ | $3520$ | $226556$ | $16966400$ | $1068504844$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $7$ | $55$ | $439$ | $4143$ | $32607$ | $261415$ | $2100679$ | $16776543$ | $134225647$ | $1073858775$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x)y=ax^5+x^4+(a^2+a+1)x^2+a^2x$
- $y^2+(x^2+x+a^2+a+1)y=a^2x^5+(a^2+1)x^3+x+a^2+a+1$
- $y^2+(x^2+x+a^2+1)y=ax^5+(a+1)x^3+x+a^2+1$
- $y^2+(x^2+x+a+1)y=(a^2+a)x^5+(a^2+a+1)x^3+x+a+1$
- $y^2+(x^2+x+a^2+a+1)y=(a^2+a)x^5+x^4+(a^2+1)x^3+x^2+1$
- $y^2+(x^2+x+a+1)y=ax^5+x^4+(a^2+a+1)x^3+x^2+1$
- $y^2+(x^2+x)y=ax^5+x^4+a^2x^3+a^2x^2+(a+1)x$
- $y^2+(x^2+x+a^2+1)y=a^2x^5+x^4+(a+1)x^3+x^2+1$
- $y^2+(x^2+x)y=(a+1)x^5+ax^3+(a^2+a+1)x^2+(a^2+a)x$
- $y^2+(x^2+x)y=a^2x^5+x^4+(a^2+a)x^3+a^2x^2+(a^2+a+1)x$
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 4.0.1025.1. |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{3}}$.
Subfield | Primitive Model |
$\F_{2}$ | 2.2.b_d |
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.8.c_ad | $2$ | 2.64.ak_cv |