Invariants
Base field: | $\F_{3}$ |
Dimension: | $3$ |
L-polynomial: | $1 - 4 x + 7 x^{2} - 10 x^{3} + 21 x^{4} - 36 x^{5} + 27 x^{6}$ |
Frobenius angles: | $\pm0.0823229705598$, $\pm0.256885878434$, $\pm0.668023839470$ |
Angle rank: | $3$ (numerical) |
Number field: | 6.0.7181504.1 |
Galois group: | $S_4\times C_2$ |
Jacobians: | $2$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $6$ | $636$ | $12978$ | $663984$ | $16071726$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $0$ | $8$ | $18$ | $100$ | $270$ | $668$ | $2184$ | $6556$ | $19440$ | $59828$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which 1 is hyperelliptic), and hence is principally polarizable:
- $y^2=2x^8+x^6+2x^4+x^3+x^2+2x+2$
- $2x^4+2x^3z+2x^2y^2+x^2yz+xy^2z+y^4+z^4=0$
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 6.0.7181504.1. |
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_h_k | $2$ | 3.9.ac_l_abo |