Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 4 x + 9 x^{2} - 36 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.116062854579$, $\pm0.586227887495$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.2873.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $51$ | $6681$ | $487152$ | $42444393$ | $3522153891$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $84$ | $666$ | $6468$ | $59646$ | $532206$ | $4782462$ | $43066884$ | $387484938$ | $3486780564$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+1)x^6+2x^4+ax^3+(a+2)x+a+2$
- $y^2=x^6+(a+1)x^4+(2a+2)x^3+x+2a$
- $y^2=2ax^6+(2a+2)x^4+(2a+2)x^3+2a$
- $y^2=(2a+2)x^6+(2a+2)x^4+(a+1)x^3+(a+1)x+2a+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{2}}$.
Endomorphism algebra over $\F_{3^{2}}$The endomorphism algebra of this simple isogeny class is 4.0.2873.1. |
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 |
---|---|---|
2.9.e_j | $2$ | 2.81.c_abt |