Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 5 x + 15 x^{2} - 45 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.125262572547$, $\pm0.528760283814$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.28749.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $47$ | $6909$ | $507647$ | $42068901$ | $3510562352$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $87$ | $695$ | $6411$ | $59450$ | $533727$ | $4784435$ | $43050003$ | $387485345$ | $3486940302$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=ax^6+(2a+1)x^4+ax^3+2x+a$
- $y^2=(2a+1)x^6+ax^4+(2a+1)x^3+2x+2a+1$
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 4.0.28749.1. |
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_p | $2$ | 2.81.f_acl |