Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 5 x + 13 x^{2} - 25 x^{3} + 25 x^{4}$ |
Frobenius angles: | $\pm0.0878807261908$, $\pm0.450170915301$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.4901.1 |
Galois group: | $D_{4}$ |
Jacobians: | $1$ |
Isomorphism classes: | 1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $9$ | $621$ | $15093$ | $353349$ | $9460944$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $1$ | $27$ | $121$ | $563$ | $3026$ | $15795$ | $78821$ | $390979$ | $1952581$ | $9771462$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+x^5+3x^3+x^2+2x+2$
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 4.0.4901.1. |
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.f_n | $2$ | 2.25.b_abf |