Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 5 x + 16 x^{2} - 45 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.146903834656$, $\pm0.519762832011$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-3}, \sqrt{-11})\) |
Galois group: | $C_2^2$ |
Jacobians: | $11$ |
Isomorphism classes: | 9 |
This isogeny class is simple but not geometrically simple, not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $48$ | $7104$ | $518400$ | $42311424$ | $3514999728$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $89$ | $710$ | $6449$ | $59525$ | $534158$ | $4785485$ | $43046369$ | $387462230$ | $3486891929$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 11 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+1)x^6+(2a+1)x^5+2x^4+ax^3+2ax^2+(a+2)x+2a+1$
- $y^2=(a+1)x^6+(a+2)x^5+(2a+1)x^3+2ax^2+2ax+a+2$
- $y^2=2ax^6+(2a+1)x^5+2x^4+(2a+2)x^3+2x^2+ax+a$
- $y^2=(2a+1)x^5+(2a+1)x^4+(2a+2)x^3+ax+2a+2$
- $y^2=2ax^6+(a+2)x^5+(a+1)x^4+x^3+(a+1)x^2+ax+a$
- $y^2=2x^6+(2a+1)x^5+(a+2)x^3+2ax^2+2ax+2a+1$
- $y^2=(a+2)x^6+2ax^5+(a+2)x^4+ax^3+2ax^2+2x+2a$
- $y^2=(2a+1)x^6+(a+2)x^5+2x^4+x^3+ax^2+x+a$
- $y^2=(a+2)x^6+(a+1)x^4+(2a+1)x^3+x^2+(a+2)x+a+2$
- $y^2=(a+2)x^6+(a+1)x^5+(a+2)x^4+x^3+x+2a+2$
- $y^2=2ax^6+(2a+1)x^5+(2a+1)x^4+2x^2+(a+2)x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{6}}$.
Endomorphism algebra over $\F_{3^{2}}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-11})\). |
The base change of $A$ to $\F_{3^{6}}$ is 1.729.ak 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-11}) \)$)$ |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{3^{2}}$.
Subfield | Primitive Model |
$\F_{3}$ | 2.3.ab_ac |
$\F_{3}$ | 2.3.b_ac |