Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $4$ |
| L-polynomial: | $( 1 - 6 x + 17 x^{2} - 30 x^{3} + 25 x^{4} )( 1 - 5 x + 13 x^{2} - 25 x^{3} + 25 x^{4} )$ |
| $1 - 11 x + 60 x^{2} - 218 x^{3} + 571 x^{4} - 1090 x^{5} + 1500 x^{6} - 1375 x^{7} + 625 x^{8}$ | |
| Frobenius angles: | $\pm0.0512862249088$, $\pm0.0878807261908$, $\pm0.384619558242$, $\pm0.450170915301$ |
| Angle rank: | $3$ (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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $63$ | $343413$ | $233700012$ | $128379112029$ | $87029776331568$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-5$ | $25$ | $121$ | $517$ | $2840$ | $15511$ | $78899$ | $391845$ | $1952581$ | $9766180$ |
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 $\times$ 2.5.af_n and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
The base change of $A$ to $\F_{5^{6}}$ is 1.15625.afm 2 $\times$ 2.15625.gn_byxl. The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{5^{2}}$
The base change of $A$ to $\F_{5^{2}}$ is 2.25.ac_av $\times$ 2.25.b_abf. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{5^{3}}$
The base change of $A$ to $\F_{5^{3}}$ is 2.125.af_dt $\times$ 2.125.a_afm. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.