Invariants
| Base field: | $\F_{3^{5}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 26 x + 243 x^{2}$ |
| Frobenius angles: | $\pm0.813926128748$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-74}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $10$ |
| Isomorphism classes: | 10 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $270$ | $58860$ | $14347530$ | $3486866400$ | $847286812350$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $270$ | $58860$ | $14347530$ | $3486866400$ | $847286812350$ | $205891158893580$ | $50031544838921370$ | $12157665459306825600$ | $2954312706607535418030$ | $717897987690317619144300$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which 0 are hyperelliptic):
- $y^2+x y=x^3+a^{148}$
- $y^2+x y=x^3+a^{202}$
- $y^2+x y=x^3+a^{56}$
- $y^2+x y=x^3+a^{180}$
- $y^2+x y=x^3+a^{168}$
- $y^2+x y=x^3+a^{130}$
- $y^2+x y=x^3+a^{122}$
- $y^2+x y=x^3+a^{60}$
- $y^2+x y=x^3+a^{20}$
- $y^2+x y=x^3+a^{124}$
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 \(\Q(\sqrt{-74}) \). |
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.aba | $2$ | (not in LMFDB) |