Invariants
Base field: | $\F_{5}$ |
Dimension: | $4$ |
L-polynomial: | $( 1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4} )^{2}$ |
$1 - 12 x + 70 x^{2} - 264 x^{3} + 699 x^{4} - 1320 x^{5} + 1750 x^{6} - 1500 x^{7} + 625 x^{8}$ | |
Frobenius angles: | $\pm0.0512862249088$, $\pm0.0512862249088$, $\pm0.384619558242$, $\pm0.384619558242$ |
Angle rank: | $1$ (numerical) |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $4$ |
Slopes: | $[0, 0, 0, 0, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $49$ | $305809$ | $239754256$ | $132002149041$ | $84618786129409$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-6$ | $22$ | $126$ | $534$ | $2754$ | $15058$ | $78282$ | $392358$ | $1953126$ | $9755062$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{6}}$.
Endomorphism algebra over $\F_{5}$The isogeny class factors as 2.5.ag_r 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{2}, \sqrt{-3})\)$)$ |
The base change of $A$ to $\F_{5^{6}}$ is 1.15625.afm 4 and its endomorphism algebra is $\mathrm{M}_{4}($\(\Q(\sqrt{-6}) \)$)$ |
- Endomorphism algebra over $\F_{5^{2}}$
The base change of $A$ to $\F_{5^{2}}$ is 2.25.ac_av 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{2}, \sqrt{-3})\)$)$ - Endomorphism algebra over $\F_{5^{3}}$
The base change of $A$ to $\F_{5^{3}}$ is 2.125.a_afm 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{2}, \sqrt{-3})\)$)$
Base change
This is a primitive isogeny class.