Invariants
Base field: | $\F_{3}$ |
Dimension: | $2$ |
L-polynomial: | $1 - x + 3 x^{2} - 3 x^{3} + 9 x^{4}$ |
Frobenius angles: | $\pm0.268536328535$, $\pm0.622727850897$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.11661.1 |
Galois group: | $D_{4}$ |
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: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $9$ | $153$ | $675$ | $8109$ | $70704$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $15$ | $27$ | $99$ | $288$ | $675$ | $2061$ | $6579$ | $19521$ | $58950$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+x^4+x^3+1$
- $y^2=x^6+x^4+x^3+2$
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 4.0.11661.1. |
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 |
---|---|---|
2.3.b_d | $2$ | 2.9.f_v |