Invariants
| Base field: | $\F_{5}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 2 x + 4 x^{2} + 10 x^{3} + 25 x^{4}$ |
| Frobenius angles: | $\pm0.380042235458$, $\pm0.803381133307$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.90944.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
This isogeny class is simple and geometrically simple, 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $42$ | $756$ | $17514$ | $414288$ | $9102282$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $30$ | $140$ | $662$ | $2908$ | $15678$ | $78184$ | $391582$ | $1955912$ | $9754350$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=4 x^5+2 x^4+2 x^2+3 x$
- $y^2=2 x^5+2 x^4+3 x^3+3 x^2+2 x+4$
- $y^2=2 x^6+x^5+4 x^4+x^3+4 x+4$
- $y^2=4 x^6+4 x^4+x^3+2 x^2+3 x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{5}$.
Endomorphism algebra over $\F_{5}$| The endomorphism algebra of this simple isogeny class is 4.0.90944.1. |
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.ac_e | $2$ | 2.25.e_ba |