Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 4 x + 9 x^{2} - 32 x^{3} + 64 x^{4}$ |
Frobenius angles: | $\pm0.110959553988$, $\pm0.574771923910$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.218768.2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $38$ | $4180$ | $237158$ | $16385600$ | $1086804598$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $67$ | $461$ | $3999$ | $33165$ | $262819$ | $2096477$ | $16787391$ | $134260733$ | $1073757507$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x+a^2+1)y=(a^2+a+1)x^5+(a+1)x^3+(a^2+1)x+a$
- $y^2+(x^2+x+a^2+a+1)y=(a+1)x^5+(a^2+1)x^3+(a^2+a+1)x+a^2$
- $y^2+(x^2+x+a+1)y=(a+1)x^5+(a^2+a+1)x^3+(a^2+a+1)x+a^2+a+1$
- $y^2+(x^2+x+a+1)y=(a^2+1)x^5+(a^2+a+1)x^3+(a+1)x+a^2+a$
- $y^2+(x^2+x+a^2+a+1)y=(a^2+a+1)x^5+(a^2+1)x^3+(a^2+1)x+a^2+1$
- $y^2+(x^2+x+a^2+1)y=(a^2+1)x^5+(a+1)x^3+(a+1)x+a+1$
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 4.0.218768.2. |
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.e_j | $2$ | 2.64.c_abv |