Invariants
Base field: | $\F_{3^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 16 x + 116 x^{2} - 432 x^{3} + 729 x^{4}$ |
Frobenius angles: | $\pm0.139208396241$, $\pm0.281528135103$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.80128.1 |
Galois group: | $D_{4}$ |
Jacobians: | $6$ |
Isomorphism classes: | 6 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $398$ | $515012$ | $390906446$ | $283283380624$ | $205985069532478$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $12$ | $706$ | $19860$ | $533046$ | $14355452$ | $387432658$ | $10460353412$ | $282429786014$ | $7625601939372$ | $205891165638466$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=ax^6+2x^5+a^2x^4+2x^3+(2a+1)x^2+(2a^2+2a+1)x+2a^2+2a+2$
- $y^2=a^2x^6+(2a+2)x^5+(2a^2+2)x^4+(a^2+2a)x^3+(a^2+2a+2)x^2+(2a+2)x+a^2+2a+2$
- $y^2=(2a^2+a+1)x^6+(2a^2+a)x^5+ax^4+(a^2+2)x^3+(a^2+1)x^2+(2a^2+2a+2)x+a^2+2a+2$
- $y^2=(2a^2+2a)x^6+(2a^2+a+2)x^5+2a^2x^4+(2a^2+2)x^3+(2a^2+a+2)x^2+(a^2+2a)x+a$
- $y^2=(a+1)x^6+(a^2+2a+2)x^5+(a^2+2a+1)x^4+(a^2+2)x^3+(a^2+a+1)x^2+(a^2+a+1)x+2a^2+1$
- $y^2=x^6+(a^2+1)x^5+x^4+(a^2+a+2)x^3+(a^2+a)x^2+(a^2+2a)x+2a^2+a+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{3}}$.
Endomorphism algebra over $\F_{3^{3}}$The endomorphism algebra of this simple isogeny class is 4.0.80128.1. |
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 |
---|---|---|
2.27.q_em | $2$ | 2.729.ay_bpy |