Invariants
Base field: | $\F_{2}$ |
Dimension: | $3$ |
L-polynomial: | $1 + 2 x^{2} + 2 x^{3} + 4 x^{4} + 8 x^{6}$ |
Frobenius angles: | $\pm0.298240499687$, $\pm0.438946264189$, $\pm0.786250280123$ |
Angle rank: | $3$ (numerical) |
Number field: | 6.0.679024.1 |
Galois group: | $S_4\times C_2$ |
Jacobians: | $2$ |
Isomorphism classes: | 2 |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $0$ |
Slopes: | $[1/3, 1/3, 1/3, 2/3, 2/3, 2/3]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $17$ | $221$ | $1139$ | $6409$ | $17357$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $9$ | $15$ | $25$ | $13$ | $69$ | $129$ | $225$ | $537$ | $1009$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which 1 is hyperelliptic), and hence is principally polarizable:
- $y^2+y=x^7+x^6+x^4$
- $x^4+x^3y+x^3z+x^2yz+xyz^2+xz^3+y^3z=0$
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 6.0.679024.1. |
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 |
---|---|---|
3.2.a_c_ac | $2$ | 3.4.e_m_bc |