Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4}$ |
Frobenius angles: | $\pm0.0512862249088$, $\pm0.384619558242$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{2}, \sqrt{-3})\) |
Galois group: | $C_2^2$ |
Jacobians: | $1$ |
Isomorphism classes: | 1 |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $7$ | $553$ | $15484$ | $363321$ | $9198847$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $24$ | $126$ | $580$ | $2940$ | $15342$ | $78204$ | $391492$ | $1953126$ | $9760344$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+4x^5+3x^4+x^3+3x^2+3x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{6}}$.
Endomorphism algebra over $\F_{5}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{2}, \sqrt{-3})\). |
The base change of $A$ to $\F_{5^{6}}$ is 1.15625.afm 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-6}) \)$)$ |
- Endomorphism algebra over $\F_{5^{2}}$
The base change of $A$ to $\F_{5^{2}}$ is the simple isogeny class 2.25.ac_av and its endomorphism algebra is \(\Q(\sqrt{2}, \sqrt{-3})\). - Endomorphism algebra over $\F_{5^{3}}$
The base change of $A$ to $\F_{5^{3}}$ is the simple isogeny class 2.125.a_afm and its endomorphism algebra is \(\Q(\sqrt{2}, \sqrt{-3})\).
Base change
This is a primitive isogeny class.