Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 3 x + 3 x^{2} - 24 x^{3} + 64 x^{4}$ |
Frobenius angles: | $\pm0.0953165727453$, $\pm0.639785697984$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.271633.1 |
Galois group: | $D_{4}$ |
Jacobians: | $3$ |
Isomorphism classes: | 3 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $41$ | $3895$ | $227468$ | $16690075$ | $1082548871$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $62$ | $441$ | $4074$ | $33036$ | $261479$ | $2098410$ | $16790866$ | $134221833$ | $1073785982$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+(a+1)x+a+1)y=x^5+x^4+a^2x^3+x+a$
- $y^2+(x^3+(a^2+a+1)x+a^2+a+1)y=x^5+x^4+ax^3+x+a^2+a$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^5+x^4+(a^2+a)x^3+x+a^2$
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 4.0.271633.1. |
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_d | $2$ | 2.64.ad_ah |