Invariants
Base field: | $\F_{5}$ |
Dimension: | $1$ |
L-polynomial: | $1 - 4 x + 5 x^{2}$ |
Frobenius angles: | $\pm0.147583617650$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-1}) \) |
Galois group: | $C_2$ |
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: | $1$ |
Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2$ | $20$ | $122$ | $640$ | $3202$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $2$ | $20$ | $122$ | $640$ | $3202$ | $15860$ | $78682$ | $391680$ | $1954562$ | $9766100$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
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 \(\Q(\sqrt{-1}) \). |
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 |
---|---|---|
1.5.e | $2$ | 1.25.ag |
1.5.ac | $4$ | 1.625.o |
1.5.c | $4$ | 1.625.o |