Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 4 x + 7 x^{2} - 32 x^{3} + 64 x^{4}$ |
Frobenius angles: | $\pm0.0429020132626$, $\pm0.591603161882$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.2873.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 6 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $36$ | $3888$ | $225612$ | $16127424$ | $1076196276$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $63$ | $437$ | $3935$ | $32845$ | $261279$ | $2092837$ | $16778815$ | $134221757$ | $1073664063$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x)y=(a+1)x^5+x^3+(a^2+1)x^2+(a^2+a+1)x$
- $y^2+(x^2+x+a+1)y=(a^2+a)x^5+x^4+x^2+a^2x+a^2$
- $y^2+(x^2+x+a^2+1)y=ax^5+x^4+x^2+(a^2+a)x+a^2+a$
- $y^2+(x^2+x+a^2+a+1)y=a^2x^5+x^4+x^2+ax+a$
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 4.0.2873.1. |
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.ab_b |
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.8.e_h | $2$ | 2.64.ac_adb |