Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 6 x + 25 x^{2} - 54 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.236852280319$, $\pm0.414859841358$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.35392.1 |
Galois group: | $D_{4}$ |
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: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $47$ | $7849$ | $586748$ | $43585497$ | $3482812847$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $96$ | $802$ | $6644$ | $58984$ | $531750$ | $4785064$ | $43041380$ | $387358066$ | $3486631536$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+1)x^6+(2a+2)x^5+(a+1)x^4+2ax^3+(2a+2)x^2+ax+2a+1$
- $y^2=(2a+1)x^6+(2a+1)x^5+2x^4+2x^3+2ax^2+ax+a$
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 4.0.35392.1. |
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_z | $2$ | 2.81.o_fj |