Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 6 x + 22 x^{2} - 54 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.162381404600$, $\pm0.459361842123$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.7600.1 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
Isomorphism classes: | 8 |
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: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $44$ | $7216$ | $545996$ | $42603264$ | $3493517324$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $90$ | $748$ | $6494$ | $59164$ | $533946$ | $4790356$ | $43047614$ | $387389332$ | $3486773850$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+2)x^6+x^5+2x^4+x^3+x^2+(2a+1)x+1$
- $y^2=ax^6+(a+1)x^5+(2a+2)x^4+(a+1)x^3+ax^2+x+2a+1$
- $y^2=ax^6+ax^5+(a+2)x^4+(2a+1)x^3+2ax^2+(a+1)x$
- $y^2=(2a+1)x^6+2ax^5+ax^4+(2a+1)x^3+(a+2)x^2+x+2a$
- $y^2=(2a+1)x^6+ax^5+ax^4+(2a+1)x^3+2ax^2+2x+2$
- $y^2=(2a+1)x^5+ax^4+ax^3+(2a+1)x+2a+1$
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 4.0.7600.1. |
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.g_w | $2$ | 2.81.i_ac |