Invariants
Base field: | $\F_{5^{2}}$ |
Dimension: | $1$ |
L-polynomial: | $1 + 5 x + 25 x^{2}$ |
Frobenius angles: | $\pm0.666666666667$ |
Angle rank: | $0$ (numerical) |
Number field: | \(\Q(\sqrt{-3}) \) |
Galois group: | $C_2$ |
Jacobians: | $2$ |
This isogeny class is simple and 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: | $[1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $31$ | $651$ | $15376$ | $391251$ | $9768751$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $31$ | $651$ | $15376$ | $391251$ | $9768751$ | $244109376$ | $6103593751$ | $152588281251$ | $3814693359376$ | $95367441406251$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{6}}$.
Endomorphism algebra over $\F_{5^{2}}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}) \). |
The base change of $A$ to $\F_{5^{6}}$ is the simple isogeny class 1.15625.ajq and its endomorphism algebra is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$. |
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 |
---|---|---|
1.25.af | $2$ | 1.625.z |
1.25.ak | $3$ | (not in LMFDB) |
1.25.k | $6$ | (not in LMFDB) |