Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 4 x + 9 x^{2} )^{2}$ |
$1 - 8 x + 34 x^{2} - 72 x^{3} + 81 x^{4}$ | |
Frobenius angles: | $\pm0.267720472801$, $\pm0.267720472801$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $3$ |
This isogeny class is not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $36$ | $7056$ | $599076$ | $45158400$ | $3514829796$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $2$ | $86$ | $818$ | $6878$ | $59522$ | $530486$ | $4774898$ | $43023038$ | $387398402$ | $3486909206$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=ax^6+x^4+(2a+1)x^3+x^2+a$
- $y^2=ax^6+(2a+2)x^4+(a+2)x^3+(2a+2)x^2+a$
- $y^2=2ax^6+(2a+2)x^4+(2a+2)x^2+2a$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{2}}$.
Endomorphism algebra over $\F_{3^{2}}$The isogeny class factors as 1.9.ae 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$ |
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.a_ae |