Invariants
Base field: | $\F_{3}$ |
Dimension: | $2$ |
L-polynomial: | $1 - x^{2} + 9 x^{4}$ |
Frobenius angles: | $\pm0.223349810481$, $\pm0.776650189519$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-5}, \sqrt{7})\) |
Galois group: | $C_2^2$ |
Jacobians: | $1$ |
Isomorphism classes: | 2 |
This isogeny class is simple but not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $9$ | $81$ | $756$ | $9801$ | $58689$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $8$ | $28$ | $116$ | $244$ | $782$ | $2188$ | $6308$ | $19684$ | $58328$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=x^6+2x^5+x^4+x^3+2x^2+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{2}}$.
Endomorphism algebra over $\F_{3}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-5}, \sqrt{7})\). |
The base change of $A$ to $\F_{3^{2}}$ is 1.9.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-35}) \)$)$ |
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 |
---|---|---|
2.3.a_b | $4$ | 2.81.bi_rj |