Invariants
| Base field: | $\F_{2^{3}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 5 x + 19 x^{2} - 40 x^{3} + 64 x^{4}$ |
| Frobenius angles: | $\pm0.224889948753$, $\pm0.460667327867$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.71825.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $39$ | $5031$ | $284076$ | $16808571$ | $1077833289$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $4$ | $78$ | $553$ | $4106$ | $32894$ | $263367$ | $2098688$ | $16766098$ | $134173849$ | $1073716878$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+(a^2+a+1)x+a^2+a+1)y=(a+1)x^3+ax+a$
- $y^2+(x^3+(a+1)x+a+1)y=x^6+(a^2+a+1)x^5+(a^2+a+1)x^4+(a+1)x^3+(a^2+1)x^2+x+a^2+a$
- $y^2+(x^3+(a+1)x+a+1)y=(a^2+1)x^3+a^2x+a^2$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+(a+1)x^5+(a+1)x^4+(a^2+1)x^3+(a^2+a+1)x^2+x+a$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^2+a+1)x^3+(a^2+a)x+a^2+a$
- $y^2+(x^3+(a^2+a+1)x+a^2+a+1)y=x^6+(a^2+1)x^5+(a^2+1)x^4+(a^2+a+1)x^3+(a+1)x^2+x+a^2$
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 4.0.71825.2. |
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_t | $2$ | 2.64.n_dl |