Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 4 x^{2} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.296115415553$, $\pm0.703884584447$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-2}, \sqrt{-5})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 18 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $54$ | $2916$ | $117126$ | $6170256$ | $282508614$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $58$ | $344$ | $2566$ | $16808$ | $116602$ | $823544$ | $5760958$ | $40353608$ | $282541978$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=3 x^6+4 x^5+6 x^4+6 x^3+3$
- $y^2=2 x^6+5 x^5+4 x^4+4 x^3+2$
- $y^2=5 x^6+5 x^5+4 x^4+5 x^3+3 x^2+5 x+4$
- $y^2=x^6+x^5+5 x^4+x^3+2 x^2+x+5$
- $y^2=3 x^6+5 x^5+3 x^4+x^3+x^2+x+1$
- $y^2=2 x^6+x^5+2 x^4+3 x^3+3 x^2+3 x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{2}}$.
Endomorphism algebra over $\F_{7}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{-5})\). |
| The base change of $A$ to $\F_{7^{2}}$ is 1.49.e 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-5}) \)$)$ |
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 |
|---|---|---|
| 2.7.a_ae | $4$ | (not in LMFDB) |
| 2.7.ag_s | $8$ | (not in LMFDB) |
| 2.7.g_s | $8$ | (not in LMFDB) |