Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 4 x + 8 x^{2} )( 1 + 8 x^{2} )$ |
$1 - 4 x + 16 x^{2} - 32 x^{3} + 64 x^{4}$ | |
Frobenius angles: | $\pm0.250000000000$, $\pm0.5$ |
Angle rank: | $0$ (numerical) |
Jacobians: | $4$ |
This isogeny class is not simple, not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $45$ | $5265$ | $279585$ | $16769025$ | $1082196225$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $81$ | $545$ | $4097$ | $33025$ | $263169$ | $2095105$ | $16760833$ | $134201345$ | $1073807361$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+y=x^5+x^4$
- $y^2+y=(a^2+a)x^5+ax^4+(a^2+a)x^3+1$
- $y^2+y=ax^5+a^2x^4+ax^3+1$
- $y^2+y=a^2x^5+a^2x^4+a^2x^3+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{24}}$.
Endomorphism algebra over $\F_{2^{3}}$The isogeny class factors as 1.8.ae $\times$ 1.8.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
The base change of $A$ to $\F_{2^{24}}$ is 1.16777216.amdc 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 1.64.a $\times$ 1.64.q. The endomorphism algebra for each factor is: - 1.64.a : \(\Q(\sqrt{-1}) \).
- 1.64.q : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- Endomorphism algebra over $\F_{2^{12}}$
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.aey $\times$ 1.4096.ey. The endomorphism algebra for each factor is: - 1.4096.aey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 1.4096.ey : the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{3}}$.
Subfield | Primitive Model |
$\F_{2}$ | 2.2.c_e |