Invariants
Base field: | $\F_{3^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 15 x + 109 x^{2} - 405 x^{3} + 729 x^{4}$ |
Frobenius angles: | $\pm0.188756140392$, $\pm0.289517143000$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.56725.1 |
Galois group: | $D_{4}$ |
Jacobians: | $3$ |
Isomorphism classes: | 3 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $419$ | $527521$ | $393675221$ | $283669670061$ | $206016005554064$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $13$ | $723$ | $19999$ | $533771$ | $14357608$ | $387429807$ | $10460258269$ | $282428922563$ | $7625596055203$ | $205891132324278$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a^2+2a)x^6+(2a^2+a+1)x^5+(a+1)x^4+(2a^2+a+1)x^3+2ax^2+(a^2+a)x+a^2+a+2$
- $y^2=2a^2x^6+(a^2+a+1)x^5+2ax^4+2x^3+(a^2+2a+2)x+a^2+a+2$
- $y^2=(a^2+2a+2)x^6+(a^2+1)x^5+(a^2+1)x^4+2x^3+2x^2+(2a^2+a)x+a$
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 4.0.56725.1. |
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.p_ef | $2$ | 2.729.ah_btt |