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 contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=ax^6+ax^3+2a$
- $y^2=(4a+3)x^6+(4a+2)x^5+(3a+2)x^4+3x^3+(3a+2)x^2+(4a+2)x+4a+3$
- $y^2=(4a+4)x^6+(2a+1)x^5+(4a+2)x^4+ax^3+ax^2+2x+2a+4$
- $y^2=(a+2)x^6+(a+1)x^5+2ax^4+3x^3+2ax^2+(a+1)x+a+2$
- $y^2=3x^6+2x^5+4x^4+(2a+4)x^3+3x^2+3x+4$
- $y^2=(4a+2)x^6+(a+3)x^5+(2a+1)x^4+(4a+1)x^3+(4a+2)x^2+(4a+2)x+2a+1$
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 |