Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 4 x + 8 x^{2} - 32 x^{3} + 64 x^{4}$ |
Frobenius angles: | $\pm0.0833333333333$, $\pm0.583333333333$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q(\zeta_{12})\) |
Galois group: | $C_2^2$ |
Jacobians: | $3$ |
This isogeny class is simple but not geometrically simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
$p$-rank: | $0$ |
Slopes: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $37$ | $4033$ | $231361$ | $16265089$ | $1082163457$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $65$ | $449$ | $3969$ | $33025$ | $262145$ | $2095105$ | $16785409$ | $134250497$ | $1073741825$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+y=(a^2+a+1)x^5+ax^4+(a^2+a+1)x^3+1$
- $y^2+y=(a^2+1)x^5+a^2x^4+(a^2+1)x^3+1$
- $y^2+y=(a+1)x^5+a^2x^4+(a+1)x^3+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{36}}$.
Endomorphism algebra over $\F_{2^{3}}$The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{12})\). |
The base change of $A$ to $\F_{2^{36}}$ is 1.68719476736.bdvoy 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$. |
- Endomorphism algebra over $\F_{2^{6}}$
The base change of $A$ to $\F_{2^{6}}$ is the simple isogeny class 2.64.a_acm and its endomorphism algebra is \(\Q(\zeta_{12})\). - Endomorphism algebra over $\F_{2^{9}}$
The base change of $A$ to $\F_{2^{9}}$ is 1.512.abg 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ - Endomorphism algebra over $\F_{2^{12}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.acm 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ - Endomorphism algebra over $\F_{2^{18}}$
The base change of $A$ to $\F_{2^{18}}$ is 1.262144.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
Base change
This is a primitive isogeny class.