Invariants
| Base field: | $\F_{3^{5}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + x + 243 x^{2}$ |
| Frobenius angles: | $\pm0.510211545821$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-971}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $15$ |
| Isomorphism classes: | 15 |
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: | $1$ |
| Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $245$ | $59535$ | $14348180$ | $3486667275$ | $847288903475$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $245$ | $59535$ | $14348180$ | $3486667275$ | $847288903475$ | $205891160262480$ | $50031544999382345$ | $12157665452311763475$ | $2954312706581785883180$ | $717897987693460711760175$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 15 curves (of which 0 are hyperelliptic):
- $y^2+x y=x^3+a^{192} x^2+a^{78}$
- $y^2+x y=x^3+a^{131} x^2+a^{116}$
- $y^2+x y=x^3+a^{127} x^2+a^{86}$
- $y^2+x y=x^3+a^{213} x^2+a^{182}$
- $y^2+x y=x^3+a^{17} x^2+a^{192}$
- $y^2+x y=x^3+a^{192} x^2+a^{88}$
- $y^2+x y=x^3+a^{127} x^2+a^{240}$
- $y^2+x y=x^3+a^{153} x^2+a^{122}$
- $y^2+x y=x^3+a^{41} x^2+a^{112}$
- $y^2+x y=x^3+a^{147} x^2+a^{160}$
- $y^2+x y=x^3+a^{33} x^2+a^{16}$
- $y^2+x y=x^3+a^{147} x^2+a^{164}$
- $y^2+x y=x^3+a^{12} x^2+a^{96}$
- $y^2+x y=x^3+a^{38} x^2+a^{38}$
- $y^2+x y=x^3+a^{185} x^2+a^{228}$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{5}}$.
Endomorphism algebra over $\F_{3^{5}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-971}) \). |
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 |
|---|---|---|
| 1.243.ab | $2$ | (not in LMFDB) |