Invariants
| Base field: | $\F_{5^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - x + 25 x^{2} )^{2}$ |
| $1 - 2 x + 51 x^{2} - 50 x^{3} + 625 x^{4}$ | |
| Frobenius angles: | $\pm0.468115719571$, $\pm0.468115719571$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $16$ |
This isogeny class is not simple, not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $625$ | $455625$ | $246490000$ | $151690775625$ | $95308846890625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $24$ | $724$ | $15774$ | $388324$ | $9759624$ | $244192174$ | $6103717224$ | $152586803524$ | $3814691138574$ | $95367452691124$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which all are hyperelliptic):
- $y^2=3 a x^6+(2 a+1) x^4+3 x+3$
- $y^2=(2 a+2) x^6+(2 a+2) x^5+(4 a+2) x^4+(4 a+2) x^3+(4 a+2) x^2+(2 a+2) x+2 a+2$
- $y^2=(3 a+4) x^6+(3 a+4) x^5+(a+1) x^4+(a+1) x^3+(a+1) x^2+(3 a+4) x+3 a+4$
- $y^2=4 x^6+(4 a+4) x^4+(2 a+3) x+4$
- $y^2=(2 a+1) x^6+(a+1) x^4+(a+2) x+2 a+1$
- $y^2=a x^6+a x^3+a$
- $y^2=(4 a+3) x^6+2 x^5+4 x^4+(4 a+4) x^3+3 x^2+3 x+2 a+4$
- $y^2=a x^6+(a+1) x^5+3 a x^4+(2 a+3) x^3+(a+4) x^2+3 x+3 a$
- $y^2=(a+2) x^6+2 x^5+4 x^4+(a+3) x^3+3 x^2+3 x+3 a+1$
- $y^2=(3 a+4) x^6+3 x^5+(a+2) x^4+(4 a+3) x^2+2 x+2 a+2$
- $y^2=(a+4) x^6+(3 a+3) x^5+(a+3) x^4+(3 a+2) x^3+(2 a+2) x^2+(3 a+4) x+2 a+3$
- $y^2=a x^6+(3 a+1) x^3+a$
- $y^2=a x^6+(4 a+4) x^5+3 a x^4+(a+2) x^3+(3 a+1) x^2+(4 a+2) x+3 a$
- $y^2=(3 a+1) x^6+(2 a+2) x^5+2 x^4+(4 a+3) x^3+(3 a+3) x^2+x+a+2$
- $y^2=3 x^6+2 x^5+4 x^4+4 x^3+2 x^2+3 x+1$
- $y^2=a x^6+(a+4) x^3+a$
where $a$ is a root of the Conway polynomial.
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.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$ |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{5^{2}}$.
| Subfield | Primitive Model |
| $\F_{5}$ | 2.5.a_ab |