Invariants
Base field: | $\F_{2}$ |
Dimension: | $4$ |
L-polynomial: | $1 - 3 x + 5 x^{2} - 7 x^{3} + 9 x^{4} - 14 x^{5} + 20 x^{6} - 24 x^{7} + 16 x^{8}$ |
Frobenius angles: | $\pm0.0368501313998$, $\pm0.278303063030$, $\pm0.465601341067$, $\pm0.738900537031$ |
Angle rank: | $4$ (numerical) |
Number field: | 8.0.8394138297.1 |
Galois group: | $C_2 \wr S_4$ |
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: | $4$ |
Slopes: | $[0, 0, 0, 0, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $3$ | $297$ | $2547$ | $58509$ | $623568$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $6$ | $6$ | $18$ | $15$ | $54$ | $126$ | $162$ | $474$ | $1101$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2 + (x^5 + x^2 + 1)y=x^{10} + x^3 + x^2 + x + 1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2}$.
Endomorphism algebra over $\F_{2}$The endomorphism algebra of this simple isogeny class is 8.0.8394138297.1. |
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 |
---|---|---|
4.2.d_f_h_j | $2$ | 4.4.b_b_ad_abb |