Invariants
Base field: | $\F_{2^{5}}$ |
Dimension: | $1$ |
L-polynomial: | $1 - 8 x + 32 x^{2}$ |
Frobenius angles: | $\pm0.250000000000$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q(\sqrt{-1}) \) |
Galois group: | $C_2$ |
Jacobians: | $1$ |
This isogeny class is simple and geometrically 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: | $[1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $25$ | $1025$ | $33025$ | $1050625$ | $33562625$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $25$ | $1025$ | $33025$ | $1050625$ | $33562625$ | $1073741825$ | $34359476225$ | $1099509530625$ | $35184363700225$ | $1125899906842625$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{20}}$.
Endomorphism algebra over $\F_{2^{5}}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1}) \). |
The base change of $A$ to $\F_{2^{20}}$ is the simple isogeny class 1.1048576.dau and its endomorphism algebra 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 the simple isogeny class 1.1024.a and its endomorphism algebra is \(\Q(\sqrt{-1}) \).
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 |
$\F_{2}$ | 1.2.c |
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
1.32.i | $2$ | 1.1024.a |
1.32.a | $8$ | (not in LMFDB) |