Invariants
Base field: | $\F_{5}$ |
Dimension: | $2$ |
L-polynomial: | $1 - x - 3 x^{2} - 5 x^{3} + 25 x^{4}$ |
Frobenius angles: | $\pm0.123439569429$, $\pm0.747770883447$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.81461.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $17$ | $493$ | $12869$ | $422501$ | $9656272$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $19$ | $101$ | $675$ | $3090$ | $15787$ | $79049$ | $390627$ | $1957601$ | $9770694$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+3x^4+4x^3+4x^2+3x+3$
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 4.0.81461.1. |
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_ad | $2$ | 2.25.ah_bx |