Invariants
Base field: | $\F_{3}$ |
Dimension: | $3$ |
L-polynomial: | $1 - 4 x + 11 x^{2} - 22 x^{3} + 33 x^{4} - 36 x^{5} + 27 x^{6}$ |
Frobenius angles: | $\pm0.132091637252$, $\pm0.376445424065$, $\pm0.544359499442$ |
Angle rank: | $3$ (numerical) |
Number field: | 6.0.5169344.1 |
Galois group: | $S_4\times C_2$ |
Jacobians: | $1$ |
Isomorphism classes: | 2 |
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: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $10$ | $1340$ | $22030$ | $455600$ | $14700050$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $16$ | $30$ | $68$ | $250$ | $772$ | $2240$ | $6812$ | $20280$ | $59036$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=2x^8+2x^7+2x^4+2x^3+2x^2+2x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3}$.
Endomorphism algebra over $\F_{3}$The endomorphism algebra of this simple isogeny class is 6.0.5169344.1. |
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 |
---|---|---|
3.3.e_l_w | $2$ | 3.9.g_l_i |