Invariants
| Base field: | $\F_{5^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 7 x + 25 x^{2} )^{2}$ |
| $1 - 14 x + 99 x^{2} - 350 x^{3} + 625 x^{4}$ | |
| Frobenius angles: | $\pm0.253183311107$, $\pm0.253183311107$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $6$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $361$ | $393129$ | $249892864$ | $153566015625$ | $95449363292761$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $628$ | $15990$ | $393124$ | $9774012$ | $244136878$ | $6103279740$ | $152586333124$ | $3814692260262$ | $95367435540628$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=a x^6+a x^3+2 a$
- $y^2=(4 a+3) x^6+(4 a+2) x^5+(3 a+2) x^4+3 x^3+(3 a+2) x^2+(4 a+2) x+4 a+3$
- $y^2=(4 a+4) x^6+(2 a+1) x^5+(4 a+2) x^4+a x^3+a x^2+2 x+2 a+4$
- $y^2=(a+2) x^6+(a+1) x^5+2 a x^4+3 x^3+2 a x^2+(a+1) x+a+2$
- $y^2=3 x^6+2 x^5+4 x^4+(2 a+4) x^3+3 x^2+3 x+4$
- $y^2=(4 a+2) x^6+(a+3) x^5+(2 a+1) x^4+(4 a+1) x^3+(4 a+2) x^2+(4 a+2) x+2 a+1$
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.ah 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$ |
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_ah |