Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 3 x + 8 x^{2} - 24 x^{3} + 64 x^{4}$ |
Frobenius angles: | $\pm0.187696427621$, $\pm0.597252504027$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.121032.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $46$ | $4600$ | $248998$ | $16974000$ | $1097431246$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $72$ | $486$ | $4144$ | $33486$ | $262824$ | $2096310$ | $16783456$ | $134213598$ | $1073615832$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+xy=ax^5+x^3+x^2+x$
- $y^2+xy=ax^5+(a^2+1)x^3+x^2+x$
- $y^2+xy=a^2x^5+x^3+x^2+x$
- $y^2+xy=a^2x^5+(a^2+a+1)x^3+x^2+x$
- $y^2+xy=(a^2+a)x^5+x^3+x^2+x$
- $y^2+xy=(a^2+a)x^5+(a+1)x^3+x^2+x$
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 4.0.121032.1. |
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_i | $2$ | 2.64.h_bw |