Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 4 x + 15 x^{2} - 32 x^{3} + 64 x^{4}$ |
Frobenius angles: | $\pm0.230612053552$, $\pm0.513287349306$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.11225.1 |
Galois group: | $D_{4}$ |
Jacobians: | $12$ |
Isomorphism classes: | 18 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $44$ | $5104$ | $273284$ | $16761536$ | $1086807964$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $79$ | $533$ | $4095$ | $33165$ | $263503$ | $2095301$ | $16762239$ | $134203709$ | $1073764239$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x+a^2+a+1)y=a^2x^5+x^4+ax^3+x^2+(a^2+1)x+a^2+a+1$
- $y^2+(x^2+x+a+1)y=ax^5+(a^2+a)x^3+ax+a^2+a+1$
- $y^2+(x^2+x)y=x^5+x^4+(a^2+1)x^3+(a^2+a)x^2+(a+1)x$
- $y^2+(x^2+x+a+1)y=(a^2+a)x^5+x^4+a^2x^3+x^2+(a^2+a+1)x+a+1$
- $y^2+(x^2+x)y=x^5+(a^2+a+1)x^3+(a^2+1)x^2+(a+1)x$
- $y^2+(x^2+x+a^2+1)y=a^2x^5+ax^3+a^2x+a+1$
- $y^2+(x^2+x+1)y=ax^5+x^4+x^2+ax+a+1$
- $y^2+(x^2+x+a^2+1)y=ax^5+x^4+(a^2+a)x^3+x^2+(a+1)x+a^2+1$
- $y^2+(x^2+x)y=x^5+x^4+(a+1)x^3+a^2x^2+(a^2+a+1)x$
- $y^2+(x^2+x+a^2+a+1)y=(a^2+a)x^5+a^2x^3+(a^2+a)x+a^2+1$
- $y^2+(x^2+x+1)y=(a^2+a)x^5+x^4+x^2+(a^2+a)x+a^2+a+1$
- $y^2+(x^2+x+1)y=a^2x^5+x^4+x^2+a^2x+a^2+1$
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 4.0.11225.1. |
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.e_p | $2$ | 2.64.o_dt |