Invariants
Base field: | $\F_{2}$ |
Dimension: | $4$ |
L-polynomial: | $1 - 3 x + 2 x^{2} + x^{4} + 8 x^{6} - 24 x^{7} + 16 x^{8}$ |
Frobenius angles: | $\pm0.0298810195513$, $\pm0.106143893905$, $\pm0.506143893905$, $\pm0.829881019551$ |
Angle rank: | $2$ (numerical) |
Number field: | 8.0.26265625.1 |
Galois group: | $C_2^2:C_4$ |
Jacobians: | $1$ |
Isomorphism classes: | 1 |
This isogeny class is simple but not geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $4$ |
Slopes: | $[0, 0, 0, 0, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1$ | $55$ | $1621$ | $27775$ | $609961$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $0$ | $0$ | $4$ | $15$ | $90$ | $105$ | $244$ | $540$ | $1075$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is not hyperelliptic), and hence is principally polarizable:
- $x^2+xy+y^2+zt=y^3+xz^2+z^3+xyt+t^3=0$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{5}}$.
Endomorphism algebra over $\F_{2}$The endomorphism algebra of this simple isogeny class is 8.0.26265625.1. |
The base change of $A$ to $\F_{2^{5}}$ is 2.32.aj_cb 2 and its endomorphism algebra is $\mathrm{M}_{2}($4.0.1025.1$)$ |
Base change
This is a primitive isogeny class.
Twists
Additional information
This is the isogeny class of the Jacobian of a function field of class number 1. This example was found by Stirpe in 2014 [10.1016/j.jnt.2014.02.016, MR:3227356], refuting a claim made in 1975 by Leitzel-Madan-Queen [0.1016/0022-314X(75)90004-9, MR:0369326].