Invariants
| Base field: | $\F_{3^{4}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 27 x + 326 x^{2} - 2187 x^{3} + 6561 x^{4}$ |
| Frobenius angles: | $\pm0.0507170516951$, $\pm0.328657711431$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.10380892.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $28$ |
| Isomorphism classes: | 28 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4674$ | $42542748$ | $282514807296$ | $1852876896297408$ | $12157198425962650194$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $55$ | $6485$ | $531604$ | $43043393$ | $3486650455$ | $282427864658$ | $22876782639031$ | $1853020197735809$ | $150094635959406916$ | $12157665463547162165$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 28 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a^3+2)x^6+2x^5+(2a^3+2a^2+2)x^4+(2a^3+2a^2+2a+2)x^3+(a^3+2a^2+1)x^2+(a^2+2a+2)x+2a^3+2a$
- $y^2=(a^3+2a^2+a+1)x^6+(2a^2+a+1)x^5+(a^3+2)x^4+(2a^3+2a^2+2a+1)x^3+(2a^3+a^2+2)x^2+(2a^3+2a^2+1)x+2a^2$
- $y^2=(2a^3+2a)x^6+2ax^5+(a^3+a)x^4+(2a^3+a+2)x^3+(2a^3+2a^2+2a+2)x^2+(a^3+a)x+2a^2+2a+2$
- $y^2=2x^6+(a^3+2a^2+2a+2)x^5+(a^3+2a+1)x^4+(a^3+2a^2+2a+2)x^3+(2a^3+2a+2)x^2+(a^2+2a)x+2a^2+a+1$
- $y^2=ax^6+(2a^2+a+2)x^5+(a^3+a^2+a+2)x^4+(a+1)x^3+a^2x^2+(2a^3+a^2+1)x+2a^3+a^2+a+1$
- $y^2=(a^3+2a^2+a)x^6+(2a^3+a^2+2)x^5+(2a^3+2a+1)x^4+x^3+(2a^3+a^2)x^2+(a+1)x+2a^3+2$
- $y^2=2ax^6+2a^2x^5+(2a^3+a^2+a)x^4+(2a^3+a^2+2a+2)x^3+ax^2+(a^3+2a+2)x+a^3+a^2+2a$
- $y^2=(a^3+a^2+a+2)x^6+x^5+(2a^3+a+2)x^4+(2a^3+a^2)x^3+(a^3+2a^2+1)x^2+x+a^3+2a^2+a+1$
- $y^2=(2a^3+a^2+1)x^6+(2a^2+2a)x^5+ax^4+(a^3+2a+2)x^3+(a^3+2a+1)x^2+(2a^3+2a+1)x+2a^3+a^2+2$
- $y^2=a^3x^6+(2a^3+2a^2+2a)x^5+2ax^4+(a^3+2a^2+a+1)x^3+(2a^3+2a+1)x^2+(a^3+a)x+2a^3+2a$
- $y^2=(2a^2+a)x^6+(2a^3+2)x^5+(a^2+a+2)x^4+(2a^3+a^2+a+1)x^3+(2a^3+2a^2+2a)x^2+(a^3+2a^2+a)x+2$
- $y^2=ax^6+(a^2+2a)x^5+(a^2+1)x^4+(2a^3+2a^2)x^3+(2a^2+2a)x^2+(2a^3+a^2)x+a^3+a^2$
- $y^2=(2a^3+a^2+a+1)x^6+(2a^3+2)x^5+2x^4+(2a^3+a^2+2a+2)x^3+(a^2+a+2)x^2+(a^2+a+2)x+2a^3+a+1$
- $y^2=(a^3+2)x^6+(a^2+a+1)x^5+(2a^2+1)x^4+2ax^3+(2a^3+2a^2+2a+2)x^2+(a^3+a^2+a+1)x+a^3+a^2+a$
- $y^2=(a^3+a^2+a+1)x^6+2a^2x^5+(2a^3+a^2+a+1)x^4+(2a^3+a)x^3+(2a^3+a^2+2a+2)x^2+(2a^3+a^2+a+2)x+2a^3+2a^2+a+1$
- $y^2=(a^3+2a+1)x^6+(a^2+2a+1)x^5+a^3x^4+(a^3+a^2+1)x^3+(a^3+2a^2+2a+1)x^2+(a^2+2a)x+2a^2+2a$
- $y^2=(a^3+a^2+2a+1)x^6+(a^3+2a^2+a+2)x^5+(2a^3+2a+1)x^4+(2a^3+2a^2+2a+1)x^3+(2a^3+2a^2+a+2)x^2+(2a^3+2a^2+1)x+2a^3+a+1$
- $y^2=(2a^3+a^2+a+2)x^6+(a^3+1)x^5+(a^2+a)x^4+x^3+(2a+1)x^2+(a^2+2a)x+a^3+a^2+2a+2$
- $y^2=(a^3+a+1)x^6+(2a^3+a^2+1)x^5+(a^2+2a+2)x^4+(a+1)x^3+(2a^3+2a^2)x^2+(2a^3+a^2+2)x+1$
- $y^2=(2a^3+a^2+a+2)x^6+(a^3+2a^2+a+1)x^5+(a^2+2a)x^4+(a^3+a+1)x^3+(a^3+2a^2+a+2)x^2+(2a^3+2a^2)x+a+2$
- $y^2=(a^3+a^2+2)x^6+a^3x^5+(a^3+a^2+a+2)x^4+(2a+2)x^3+(a^2+2a+2)x^2+(a^3+2a^2+2)x+a^3+a^2+2$
- $y^2=(2a^3+2a^2+2a+2)x^6+(a^2+a)x^5+(2a^3+a)x^4+2a^3x^3+(a^3+2a+2)x^2+(a^3+2a^2+a)x+a^3+a^2+2a+2$
- $y^2=2ax^6+(a^3+2a+2)x^5+(a^3+2)x^4+(a^3+a^2+2a)x^3+2x^2+(a^3+a)x+2a+1$
- $y^2=(a^3+2a+2)x^6+2a^3x^5+(2a^2+2a+2)x^4+ax^3+(a^2+2a+2)x^2+(a^3+2a+2)x+2a^2+2a$
- $y^2=(a^2+2)x^6+(2a^3+a+1)x^5+(a^3+2a^2+a+2)x^4+(2a^2+2)x^3+(2a+2)x^2+(a^3+a^2+2a+1)x+a^3+a+2$
- $y^2=(a^3+2a^2+a+1)x^6+(a^2+a+2)x^5+(a^3+a^2+2a+2)x^4+(2a^3+a+1)x^3+(2a^3+2a^2+a)x^2+(2a^2+1)x+a^3+2a+2$
- $y^2=(2a^3+a^2+2)x^6+(2a^2+2a)x^5+(a^3+2a^2+2a)x^4+(2a^3+2a^2)x^2+(a^2+a+1)x+a^3+a+1$
- $y^2=2x^6+(2a^3+2a^2+2a+2)x^5+(2a^3+2a^2+1)x^4+(2a^3+a^2)x^2+(2a^3+2a+2)x+2a^3+2a^2+a+2$
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 4.0.10380892.1. |
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 |
|---|---|---|
| 2.81.bb_mo | $2$ | (not in LMFDB) |