Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $1 - x - 5 x^{3} + 25 x^{4}$ |
Frobenius angles: | $\pm0.189652078905$, $\pm0.706462326177$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.8405.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
This isogeny class is simple and geometrically 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $20$ | $640$ | $13760$ | $442880$ | $10080100$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $25$ | $110$ | $705$ | $3225$ | $15670$ | $78965$ | $389985$ | $1950230$ | $9765825$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^5+4x^2+1$
- $y^2=x^5+4x^2+2$
- $y^2=4x^6+2x^5+3x^3+4x+2$
- $y^2=2x^6+2x^4+4x^3+x+1$
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 4.0.8405.1. |
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.b_a | $2$ | 2.25.ab_bo |