Invariants
| Base field: | $\F_{3^{4}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 27 x + 331 x^{2} - 2187 x^{3} + 6561 x^{4}$ |
| Frobenius angles: | $\pm0.0987890015234$, $\pm0.315475092415$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.19250077.2 |
| Galois group: | $D_{4}$ |
| Jacobians: | $24$ |
| Isomorphism classes: | 24 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4679$ | $42611653$ | $282730675871$ | $1853221738768533$ | $12157558519443173744$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $55$ | $6495$ | $532009$ | $43051403$ | $3486753730$ | $282428838543$ | $22876790280301$ | $1853020263362099$ | $150094636703435251$ | $12157665472191003390$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a^3+2a^2)x^6+(2a^3+2a+1)x^5+(a^2+2)x^4+(a^3+2a)x^3+(a^3+2a^2)x^2+(2a^3+2)x+a^3+1$
- $y^2=(a^3+a+2)x^6+(2a^2+a+1)x^5+(a^3+2a^2+2a+1)x^4+(2a^3+a^2+2a+1)x^3+(2a^3+2a^2+a+1)x^2+(2a^3+a^2)x+a^3+a^2+a$
- $y^2=(a^3+a)x^6+(a^3+2a^2+2)x^5+(2a^3+a+1)x^4+(2a^3+a+2)x^3+a^3x^2+(2a^3+2a)x+2a^3+2a^2+2a+1$
- $y^2=(a^2+a+2)x^6+(2a^3+2a+1)x^5+(2a^2+2a+2)x^4+(a^2+a+2)x^3+(2a^3+a^2+1)x^2+(a^2+1)x+a^2+2a$
- $y^2=(a^3+2a^2)x^6+(2a^2+2a)x^5+2x^4+(2a^3+2a^2)x^3+(2a^3+2)x^2+(a^3+2a^2+a+2)x+2a^3+2a^2+2a$
- $y^2=(a+1)x^6+(a+2)x^5+2a^2x^4+(2a^3+a^2)x^3+(a^3+a^2+2)x^2+(a^3+a^2+2a+1)x+2a^2+a+2$
- $y^2=(a^3+2a^2)x^6+(a^3+2a^2+2a)x^5+(a^3+2a)x^4+(a^3+a^2+a+1)x^3+(2a^3+2a^2+2a+2)x^2+(a^3+2a^2)x+a^3+a^2+2a$
- $y^2=(a^3+a^2+2a+2)x^6+(2a^3+2)x^5+(a^3+a^2+a+2)x^4+(a^3+a^2+2)x^3+(a^2+2a+1)x^2+(2a^3+a^2+1)x+2a+1$
- $y^2=2a^3x^6+(2a^3+2a^2+2a+2)x^5+(a^3+1)x^4+(2a^3+2a^2)x^3+(2a^2+1)x^2+(a^3+a)x+a^3+a^2+2$
- $y^2=(a+2)x^6+(2a^3+a+2)x^5+(2a^3+2a)x^4+(a^3+2a^2+a+1)x^3+(a^3+a^2+2)x^2+(2a^2+1)x+2a^3+a^2+2a$
- $y^2=(2a^3+a^2+2)x^6+(2a^3+2a^2+2a+2)x^5+x^4+(2a^3+a^2+a+2)x^3+2a^2x^2+(2a^2+2)x+a+1$
- $y^2=(2a^3+2a^2+2a+2)x^6+(2a^3+a+2)x^5+(a^3+2a^2+2a+1)x^4+(2a^3+2a+1)x^3+(a^3+2a^2+2)x^2+(2a^3+a+1)x+a$
- $y^2=(a^3+2a^2+a+1)x^6+(2a^3+2a^2+2a)x^5+(a^2+2a+2)x^4+(a^3+a+1)x^3+(a^3+2a^2+2a)x^2+(2a^3+2a^2+a+1)x+2a+1$
- $y^2=(2a^3+a+1)x^6+(2a^3+a^2+2a+2)x^5+(2a^2+2a+2)x^4+(a^3+2a^2+a)x^3+(2a^3+2a^2+2a)x^2+2x+2a^3+a+1$
- $y^2=(a^2+a)x^6+2ax^5+(a^2+1)x^4+(2a^3+2a^2+a)x^3+a^3x^2+(2a^2+a+1)x+2a^3+2a$
- $y^2=(a^3+a^2+a+1)x^6+2a^2x^5+2ax^4+(2a^3+a+2)x^3+(a^3+a^2+a+2)x^2+(a^3+a+2)x+a^3+2a^2+a+2$
- $y^2=ax^6+a^2x^5+(a^3+a+2)x^4+(2a^3+2a^2+a+2)x^3+(a^2+2a+1)x^2+(a^3+a^2+2a+1)x+2a^3+a^2$
- $y^2=2a^2x^6+(a^3+a^2+a)x^5+2ax^4+(a^3+a^2+a+2)x^3+(2a^3+2a+2)x^2+(2a^3+2a^2+2a+2)x+2a^3+a^2+a+2$
- $y^2=(2a^3+a^2+2a+1)x^6+(a^2+a+1)x^5+(a^3+2a^2+2a+1)x^4+(2a^3+2a)x^3+(2a^2+2)x^2+(2a^3+a^2+2)x+a^3+a^2+a$
- $y^2=(a^3+2a)x^6+2x^5+a^3x^4+(a^2+a)x^3+(a^2+2a+2)x^2+(a^3+2a^2+a+1)x+2a^3+a^2+a+1$
- $y^2=(a^2+1)x^6+(2a^3+2a^2+a)x^5+(a^3+a^2+a+2)x^4+2x^3+(2a^3+2a^2+2a+1)x^2+2ax+2a^3+a^2+2a+2$
- $y^2=(2a^3+a^2+a+1)x^6+(a^3+a^2+a+2)x^5+(a^3+a^2+2a+2)x^4+(a^3+2a^2)x^3+(2a^2+2a+2)x^2+(a^3+1)x+a^3+2a^2$
- $y^2=(2a^3+a+2)x^6+(2a^2+2a+2)x^5+(a^3+2)x^4+(2a^3+a^2+a+2)x^3+(a^3+2a^2+a+1)x^2+(2a^3+2a+1)x+2a^2+2a+2$
- $y^2=(a^3+a^2+2a+2)x^6+(2a^3+2a)x^5+(2a^2+a)x^4+2ax^3+(2a^2+a+2)x^2+(a^3+2a^2+a+1)x+a^3+a^2+a+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{4}}$.
Endomorphism algebra over $\F_{3^{4}}$| The endomorphism algebra of this simple isogeny class is 4.0.19250077.2. |
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 |
|---|---|---|
| 2.81.bb_mt | $2$ | (not in LMFDB) |