Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 5 x + 23 x^{2} - 45 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.293969626750$, $\pm0.426020201528$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.19525.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $55$ | $8525$ | $596695$ | $43315525$ | $3463262000$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $103$ | $815$ | $6603$ | $58650$ | $530623$ | $4783035$ | $43044563$ | $387403745$ | $3486819598$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+1)x^6+(2a+2)x^5+2x^4+(a+2)x^3+ax^2+(2a+2)x$
- $y^2=(a+2)x^6+x^5+(a+1)x^4+(2a+1)x^3+ax^2+x$
- $y^2=2ax^6+x^5+ax^4+(a+2)x^2+ax+2a$
- $y^2=(a+2)x^6+(2a+2)x^5+(a+1)x^4+2ax^3+2ax^2+(2a+2)x+1$
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 4.0.19525.1. |
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_x | $2$ | 2.81.v_jh |