Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 5 x + 13 x^{2} - 45 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.0703393266913$, $\pm0.545465958288$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1525.1 |
Galois group: | $D_{4}$ |
Jacobians: | $5$ |
Isomorphism classes: | 7 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $45$ | $6525$ | $486405$ | $41505525$ | $3492738000$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $83$ | $665$ | $6323$ | $59150$ | $532043$ | $4779185$ | $43043843$ | $387474245$ | $3486871598$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 5 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=ax^6+ax^5+2ax$
- $y^2=ax^6+2ax^5+(a+1)x^4+(a+2)x^3+2ax+1$
- $y^2=x^6+(2a+1)x^5+(2a+2)x^4+(2a+1)x^3+(2a+2)x^2+2ax+a$
- $y^2=ax^6+(a+1)x^5+2ax^4+x^3+2x^2+(a+1)x+2a+1$
- $y^2=ax^6+2ax^5+2x^4+(2a+1)x^3+2ax+2a+2$
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 4.0.1525.1. |
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.f_n | $2$ | 2.81.b_aep |