Invariants
Base field: | $\F_{5^{4}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 25 x )^{4}$ |
$1 - 100 x + 3750 x^{2} - 62500 x^{3} + 390625 x^{4}$ | |
Frobenius angles: | $0$, $0$, $0$, $0$ |
Angle rank: | $0$ (numerical) |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $331776$ | $151613669376$ | $59589387451109376$ | $23282825947723387109376$ | $9094943292439556121787109376$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $526$ | $388126$ | $244078126$ | $152586328126$ | $95367392578126$ | $59604643798828126$ | $37252902960205078126$ | $23283064364776611328126$ | $14551915228351593017578126$ | $9094947017728900909423828126$ |
Jacobians and polarizations
This isogeny class contains a Jacobian, and hence is principally polarizable.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{4}}$.
Endomorphism algebra over $\F_{5^{4}}$The isogeny class factors as 1.625.aby 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $5$ and $\infty$. |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{5^{4}}$.
Subfield | Primitive Model |
$\F_{5}$ | 2.5.a_ak |
$\F_{5}$ | 2.5.a_k |