Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 4 x + 16 x^{2} - 36 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.234075997255$, $\pm0.523868533114$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.334080.4 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $58$ | $8004$ | $547114$ | $43061520$ | $3522170698$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $98$ | $750$ | $6566$ | $59646$ | $533186$ | $4778934$ | $43023806$ | $387407910$ | $3486814178$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+2ax^5+(a+1)x^4+(a+1)x^2+(2a+2)x+2a+1$
- $y^2=(a+2)x^6+2ax^4+(2a+2)x^3+2ax^2+x+a+2$
- $y^2=(2a+1)x^6+(a+2)x^5+(a+1)x^4+(a+1)x^3+2ax+2a$
- $y^2=(2a+2)x^6+2ax^4+(2a+2)x^3+(a+2)x^2+x+a+2$
- $y^2=(a+1)x^6+(a+2)x^4+(a+1)x^3+2ax^2+x+2a$
- $y^2=x^6+(2a+2)x^5+2ax^4+(a+2)x^3+x^2+ax+a$
- $y^2=2ax^6+(2a+2)x^5+2ax^4+x^3+(2a+2)x^2+(a+2)x+a+2$
- $y^2=(2a+2)x^6+x^5+2ax^4+(2a+1)x^3+(2a+2)x^2+ax+a$
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 4.0.334080.4. |
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.e_q | $2$ | 2.81.q_fa |