Invariants
Base field: | $\F_{2^{5}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 8 x + 32 x^{2} )^{2}$ |
$1 - 16 x + 128 x^{2} - 512 x^{3} + 1024 x^{4}$ | |
Frobenius angles: | $\pm0.250000000000$, $\pm0.250000000000$ |
Angle rank: | $0$ (numerical) |
Jacobians: | $5$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $625$ | $1050625$ | $1090650625$ | $1103812890625$ | $1126449796890625$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $17$ | $1025$ | $33281$ | $1052673$ | $33570817$ | $1073741825$ | $34359214081$ | $1099507433473$ | $35184355311617$ | $1125899906842625$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+y=(a+1)x^5+x^4+(a+1)x^3$
- $y^2+y=(a^3+a^2)x^5+(a^2+a)x^4+(a^3+a^2)x^3+1$
- $y^2+y=(a^2+1)x^5+ax^4+(a^2+1)x^3+1$
- $y^2+y=(a^4+a^3+a)x^5+x^4+(a^4+a^3+a)x^3$
- $y^2+y=(a^4+1)x^5+a^2x^4+(a^4+1)x^3+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{20}}$.
Endomorphism algebra over $\F_{2^{5}}$The isogeny class factors as 1.32.ai 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
The base change of $A$ to $\F_{2^{20}}$ is 1.1048576.dau 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 $\F_{2^{10}}$
The base change of $A$ to $\F_{2^{10}}$ is 1.1024.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{5}}$.
Subfield | Primitive Model |
$\F_{2}$ | 2.2.e_i |