Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 3 x + 7 x^{2} - 15 x^{3} + 25 x^{4}$ |
Frobenius angles: | $\pm0.177952114464$, $\pm0.556618995437$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.48069.2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $15$ | $765$ | $14715$ | $386325$ | $10381200$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $31$ | $117$ | $619$ | $3318$ | $15991$ | $77997$ | $390739$ | $1955043$ | $9759526$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+x^5+4x^4+3x+4$
- $y^2=3x^6+3x^5+2x^4+3x^3+3x+2$
- $y^2=4x^6+x^5+2x^3+4x^2+4x+2$
- $y^2=x^6+3x^5+4x^4+4x^3+3x+3$
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 4.0.48069.2. |
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.d_h | $2$ | 2.25.f_j |