Invariants
| Base field: | $\F_{5^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 5 x )^{2}( 1 - 2 x + 25 x^{2} )$ |
| $1 - 12 x + 70 x^{2} - 300 x^{3} + 625 x^{4}$ | |
| Frobenius angles: | $0$, $0$, $\pm0.435905783151$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $4$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
| $p$-rank: | $1$ |
| Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $384$ | $387072$ | $242448768$ | $151763189760$ | $95254866985344$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $14$ | $622$ | $15518$ | $388510$ | $9754094$ | $244120462$ | $6103513598$ | $152587140670$ | $3814689566414$ | $95367403741102$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+4)x^5+(2a+1)x^4+(3a+1)x^3+(2a+1)x^2+(2a+4)x$
- $y^2=(3a+1)x^6+(3a+4)x^5+4ax^4+(3a+2)x^3+4ax^2+(3a+4)x+3a+1$
- $y^2=(2a+4)x^6+(4a+4)x^5+(2a+4)x^4+(3a+3)x^3+(2a+4)x^2+(4a+4)x+2a+4$
- $y^2=(3a+2)x^6+(3a+4)x^5+(4a+3)x^4+4ax^3+(4a+3)x^2+(3a+4)x+3a+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{2}}$.
Endomorphism algebra over $\F_{5^{2}}$The isogeny class factors as 1.25.ak $\times$ 1.25.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.