Invariants
| Base field: | $\F_{3^{3}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 15 x + 103 x^{2} - 405 x^{3} + 729 x^{4}$ |
| Frobenius angles: | $\pm0.0625058430386$, $\pm0.346918783446$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.293509.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $413$ | $517489$ | $388280711$ | $282150009981$ | $205757737375568$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $13$ | $711$ | $19729$ | $530915$ | $14339608$ | $387367347$ | $10460267719$ | $282430345619$ | $7625603683783$ | $205891145855286$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a^2+2a+1)x^6+(2a+1)x^5+2x^4+(a^2+2a+1)x^3+(2a^2+a+1)x^2+(2a^2+2a+2)x+a^2+a+2$
- $y^2=(2a^2+a)x^6+(a^2+1)x^5+(a^2+2a+1)x^4+(a+1)x^2+a^2x+2a^2+2a+2$
- $y^2=a^2x^6+(2a+2)x^5+2x^4+a^2x^3+(2a^2+2)x^2+(2a^2+a+2)x+a^2+2a+2$
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 4.0.293509.1. |
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_dz | $2$ | 2.729.at_adf |