Invariants
Base field: | $\F_{3^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 14 x + 97 x^{2} - 378 x^{3} + 729 x^{4}$ |
Frobenius angles: | $\pm0.136634956619$, $\pm0.355731932508$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.940608.1 |
Galois group: | $D_{4}$ |
Jacobians: | $12$ |
Isomorphism classes: | 12 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $435$ | $530265$ | $391296420$ | $282729344025$ | $205866829758675$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $14$ | $728$ | $19880$ | $532004$ | $14347214$ | $387417806$ | $10460549498$ | $282431494916$ | $7625606082920$ | $205891138653368$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+2)x^6+2ax^5+(a^2+2a)x^4+(a^2+a+1)x^3+2ax^2+ax+a^2+a+1$
- $y^2=(a+1)x^6+(2a+1)x^5+(2a^2+2a+1)x^4+(2a^2+2a)x^3+(a^2+a+1)x^2+(2a^2+2a)x+1$
- $y^2=(2a^2+2a)x^6+(2a^2+2a)x^5+a^2x^4+(2a^2+2a+2)x^3+(2a^2+2)x^2+(a^2+a)x+2a^2+a$
- $y^2=(2a^2+2a)x^6+(a+1)x^5+a^2x^4+a^2x^3+(a^2+a+1)x+2a^2$
- $y^2=(a^2+a+2)x^6+(a^2+2a+2)x^5+(2a^2+a+1)x^4+(a^2+a)x^3+(a^2+a+2)x^2+(2a^2+a+1)x+a^2+a+2$
- $y^2=x^6+(2a^2+a+2)x^5+(a^2+a)x^4+(a^2+a)x^3+(2a+1)x^2+a$
- $y^2=(a^2+2a)x^6+(2a^2+a)x^5+2x^4+(a+1)x^3+(2a^2+1)x^2+(2a^2+2a+1)x+2$
- $y^2=(a^2+a+2)x^6+(a^2+a+2)x^5+(a+1)x^4+(2a^2+2a+1)x^3+(2a^2+a+1)x^2+(2a^2+a+2)x+a^2$
- $y^2=(a^2+2a+2)x^6+(2a+1)x^5+(2a^2+a+2)x^4+2ax^3+(a^2+2)x^2+(2a^2+2a+2)x+a^2+2a+2$
- $y^2=(2a^2+a+1)x^6+(2a+2)x^5+(a^2+a)x^4+(a^2+a)x^3+ax^2+(a^2+2a+2)x+a^2+2a+1$
- $y^2=(a^2+1)x^6+(a^2+a)x^5+2x^4+(2a+2)x^3+(2a^2+1)x+2a$
- $y^2=(2a^2+a+2)x^6+(2a+1)x^5+2a^2x^4+(2a^2+2)x^3+2a^2x+2a+1$
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 4.0.940608.1. |
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.o_dt | $2$ | 2.729.ac_kx |