Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 5 x + 13 x^{2} - 40 x^{3} + 64 x^{4}$ |
Frobenius angles: | $\pm0.0644257339289$, $\pm0.530510095142$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.56129.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $33$ | $4059$ | $237996$ | $15947811$ | $1072956093$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $66$ | $463$ | $3890$ | $32744$ | $262647$ | $2094908$ | $16771714$ | $134241079$ | $1073808186$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+(a+1)x+a+1)y=x^6+(a^2+1)x^5+(a^2+1)x^4+(a^2+a+1)x^3+(a^2+1)x^2+x$
- $y^2+(x^3+(a^2+a+1)x+a^2+a+1)y=x^6+(a+1)x^5+(a+1)x^4+(a^2+1)x^3+(a+1)x^2+x$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+(a^2+a+1)x^5+(a^2+a+1)x^4+(a+1)x^3+(a^2+a+1)x^2+x$
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 4.0.56129.1. |
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.f_n | $2$ | 2.64.b_adz |