Invariants
| Base field: | $\F_{3^{4}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 29 x + 368 x^{2} - 2349 x^{3} + 6561 x^{4}$ |
| Frobenius angles: | $\pm0.128118350588$, $\pm0.257159505691$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.608056.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4552$ | $42370016$ | $282738266752$ | $1853604985288576$ | $12158098148309661512$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $53$ | $6457$ | $532022$ | $43060305$ | $3486908493$ | $282430187902$ | $22876793781741$ | $1853020189977569$ | $150094635557415062$ | $12157665465037360777$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a^3+2a^2+2a+1)x^6+(2a^3+2a+1)x^5+(2a^2+a+1)x^4+(a^3+1)x^3+(a^3+2)x^2+(a^3+a^2+a+2)x+2a^3+2a^2$
- $y^2=2a^3x^6+(2a^3+2)x^5+(2a^3+2a^2+a+1)x^4+(2a^2+1)x^3+(2a^2+a)x^2+(a^3+a^2+1)x+a^3+a+1$
- $y^2=(2a^3+2a^2+2)x^6+(2a^3+a^2+a+1)x^5+(2a^3+2)x^4+(a^3+2a+2)x^3+(a^3+a+1)x^2+(a^3+2a+1)x+a^3+2a^2+2a$
- $y^2=(2a^3+a^2+2)x^6+(a^3+a^2+a)x^5+(2a^3+a^2+2a+2)x^4+(a^3+2a^2+1)x^3+(2a^3+2a^2+a+2)x^2+ax+a^3+a^2+2$
- $y^2=(a^3+2a^2)x^6+(2a^3+2a+2)x^5+(2a^2+2)x^4+(2a^3+2a^2+2a+2)x^3+(2a^3+a^2+a+1)x^2+(2a^3+2a+1)x+2a^3+2a^2+a+2$
- $y^2=(a^3+2a^2+2a+2)x^6+(a^3+a^2+a+1)x^5+(2a^3+a^2+2a)x^4+2a^3x^3+(2a^3+a^2+a+1)x^2+(2a^3+2a^2)x+2a^3+a+1$
- $y^2=(2a^3+a^2+a+1)x^6+ax^5+(a^3+a^2+2)x^4+(2a^3+2)x^3+(a^3+2a^2)x^2+(2a+2)x+2a^3+1$
- $y^2=(a^3+2a+2)x^6+(2a^3+2a^2+2)x^5+(a^2+1)x^4+(2a^3+a+2)x^3+2a^3x^2+(a^3+a)x+a^3$
- $y^2=(2a^3+a^2+2a+1)x^6+(2a^3+a^2+1)x^5+(2a^2+a+1)x^4+(2a^2+2)x^3+(a^3+a^2+1)x^2+(2a^3+2a^2+2a+1)x+a^2+2$
- $y^2=(a^3+a^2+1)x^6+(2a^3+a+1)x^5+(2a^2+2a+2)x^4+(a^3+2a^2)x^3+(2a^3+a+2)x^2+(2a^3+a^2+2a+2)x+2a^2+a+1$
- $y^2=(a^3+a^2)x^6+(2a^2+2)x^5+(2a^3+2a^2+2a)x^4+(2a^3+2a+1)x^3+2ax^2+(a^3+a^2+1)x+2a^3+1$
- $y^2=(a^3+2a^2)x^6+(2a^2+2a)x^5+(a^3+2a^2+1)x^4+(2a^3+a^2)x^3+(a^2+2a)x^2+(a^3+a^2+2)x+a^3+a^2+a+2$
- $y^2=(a^3+2a^2+1)x^6+(a^3+a^2+2)x^5+(a^2+a)x^4+(a^3+a^2)x^3+(2a^2+2a)x^2+(2a^3+a^2+2)x+2a^2+2a$
- $y^2=(2a^3+a^2+2)x^6+(a^2+2a+2)x^5+(2a^3+2a^2+2)x^4+(2a+2)x^3+(a^3+2a^2)x^2+(a+2)x+2a^3+2a+2$
- $y^2=2a^2x^6+a^2x^5+(a^3+2a^2+a+1)x^4+(a^3+a^2+a)x^3+(a^2+1)x^2+(a^3+2a^2+2)x+a+1$
- $y^2=(a^3+2a^2+2a+2)x^6+(2a^3+a+1)x^5+(2a^3+a+2)x^4+(a^2+2a)x^3+(2a^3+a^2+a+1)x^2+(2a^3+a^2)x+a^3+2a^2+a+2$
- $y^2=(a^3+2a^2+2a+2)x^6+(2a^3+2a^2+a)x^5+(a^3+a+1)x^4+(2a^3+2a)x^3+(a+1)x^2+(2a^3+2a^2+a)x+2a^2+2a$
- $y^2=(2a^3+2a+1)x^6+(2a^3+1)x^5+(2a^2+1)x^4+(a^2+2a+1)x^3+(a^2+2a+2)x^2+(a^3+2a^2+2a+1)x+a^3+a^2+2a+2$
- $y^2=(a^3+a^2+2a)x^6+2a^3x^5+(2a^3+a)x^4+(a^2+2)x^3+2a^3x^2+(a^2+a+2)x+a^3+2a+1$
- $y^2=(2a^3+2)x^6+ax^5+(2a^2+2a)x^4+(a^3+2a^2)x^3+(a^3+2a+2)x^2+a^2x+a+2$
- $y^2=ax^6+(2a^3+1)x^5+(2a^3+a^2+2a+2)x^4+(2a^3+a^2+a+1)x^3+(a^2+2a+2)x^2+(2a^3+2a^2)x+a^3+2a+2$
- $y^2=(2a+2)x^6+(a^2+2a+1)x^5+(2a^3+a)x^4+(2a^3+a^2)x^3+(a^3+a^2+a+2)x^2+(a^3+2a^2+2a+1)x+a^3+a^2+2a$
- $y^2=(2a^3+a^2)x^6+(2a^3+a^2+2a)x^5+(a^3+2a+1)x^4+(2a^3+2a^2+a+1)x^3+(a^3+a^2+1)x^2+(a^3+a^2+2)x+a^2+2a$
- $y^2=(a^3+a^2+2a+1)x^6+(a^3+a+1)x^5+(a^3+a+1)x^4+(a^2+2a+2)x^3+(a^3+a^2+2a+1)x^2+(2a^3+2a^2+2a+1)x+a^2+a+1$
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 4.0.608056.1. |
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.bd_oe | $2$ | (not in LMFDB) |