Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 4 x + 5 x^{2} )( 1 + 5 x^{2} )$ |
| $1 - 4 x + 10 x^{2} - 20 x^{3} + 25 x^{4}$ | |
| Frobenius angles: | $\pm0.147583617650$, $\pm0.5$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $4$ |
| Isomorphism classes: | 12 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $12$ | $720$ | $15372$ | $368640$ | $10009452$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $2$ | $30$ | $122$ | $590$ | $3202$ | $16110$ | $78682$ | $390430$ | $1954562$ | $9772350$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=x^5+2x^2+2$
- $y^2=2x^6+x^5+2x^4+3x^3+3x^2+x+3$
- $y^2=3x^6+3x^5+3x^4+3x^2+3x+3$
- $y^2=3x^6+2x^5+3x^4+3x^3+4x^2+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5^{2}}$.
Endomorphism algebra over $\F_{5}$| The isogeny class factors as 1.5.ae $\times$ 1.5.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
The base change of $A$ to $\F_{5^{2}}$ is 1.25.ag $\times$ 1.25.k. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.5.e_k | $2$ | 2.25.e_ak |
| 2.5.ac_k | $4$ | 2.625.abk_ve |
| 2.5.c_k | $4$ | 2.625.abk_ve |