Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 5 x + 19 x^{2} - 45 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.205601850040$, $\pm0.488925242199$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.206829.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $51$ | $7701$ | $551259$ | $42886869$ | $3510481776$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $95$ | $755$ | $6539$ | $59450$ | $533951$ | $4784855$ | $43028339$ | $387373265$ | $3486779150$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+1)x^6+(a+1)x^5+x^3+(2a+1)x^2+2x+2a$
- $y^2=(2a+1)x^6+2ax^5+(2a+2)x^4+x^3+x^2+2a+2$
- $y^2=(a+1)x^6+2x^4+2x^3+(a+1)x^2+2x+a$
- $y^2=ax^6+(2a+2)x^5+(a+2)x^4+x^3+2x+a$
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 4.0.206829.1. |
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_t | $2$ | 2.81.n_cv |