Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 3 x + 4 x^{2} - 24 x^{3} + 64 x^{4}$ |
Frobenius angles: | $\pm0.117638066933$, $\pm0.631737784799$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.90972.1 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $42$ | $4032$ | $231714$ | $16781184$ | $1087503522$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $64$ | $450$ | $4096$ | $33186$ | $261952$ | $2098914$ | $16793344$ | $134227746$ | $1073761984$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+xy=ax^5+(a^2+a)x^3+x^2+x$
- $y^2+xy=a^2x^5+ax^3+x^2+x$
- $y^2+xy=(a+1)x^5+ax^3+x^2+x$
- $y^2+xy=(a^2+a)x^5+a^2x^3+x^2+x$
- $y^2+xy=(a^2+a+1)x^5+(a^2+a)x^3+x^2+x$
- $y^2+xy=(a^2+1)x^5+a^2x^3+x^2+x$
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 4.0.90972.1. |
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_e | $2$ | 2.64.ab_a |