Invariants
| Base field: | $\F_{5^{2}}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 8 x + 25 x^{2} )^{2}$ |
| $1 - 16 x + 114 x^{2} - 400 x^{3} + 625 x^{4}$ | |
| Frobenius angles: | $\pm0.204832764699$, $\pm0.204832764699$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $5$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $324$ | $374544$ | $246929796$ | $153413222400$ | $95489208772164$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $598$ | $15802$ | $392734$ | $9778090$ | $244187638$ | $6103580122$ | $152587231294$ | $3814690378570$ | $95367393027478$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=(a+3)x^6+3x^4+3x^2+a+3$
- $y^2=x^6+(a+1)x^3+3$
- $y^2=x^6+(4a+2)x^3+3$
- $y^2=(3a+2)x^6+(4a+4)x^5+3x^4+3ax^3+2x^2+(4a+4)x+2a+3$
- $y^2=(4a+3)x^6+(a+1)x^5+(4a+2)x^4+3ax^3+(a+3)x^2+(a+1)x+a+2$
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.ai 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
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_ai |