Invariants
| Base field: | $\F_{3}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 2 x + 2 x^{2} - 6 x^{3} + 9 x^{4}$ |
| Frobenius angles: | $\pm0.116139763599$, $\pm0.616139763599$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{5})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $2$ |
| Isomorphism classes: | 3 |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4$ | $80$ | $436$ | $6400$ | $69044$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $2$ | $10$ | $14$ | $78$ | $282$ | $730$ | $2214$ | $6878$ | $19922$ | $59050$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^5+x^4+x^3+x^2+x+2$
- $y^2=2x^6+2x^5+x^4+x^2+x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{4}}$.
Endomorphism algebra over $\F_{3}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{5})\). |
| The base change of $A$ to $\F_{3^{4}}$ is 1.81.ac 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$ |
- Endomorphism algebra over $\F_{3^{2}}$
The base change of $A$ to $\F_{3^{2}}$ is the simple isogeny class 2.9.a_ac and its endomorphism algebra is \(\Q(i, \sqrt{5})\).
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 |
|---|---|---|
| 2.3.c_c | $2$ | 2.9.a_ac |
| 2.3.a_ae | $8$ | (not in LMFDB) |
| 2.3.a_e | $8$ | (not in LMFDB) |