Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 4 x + 8 x^{2} + 28 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.429508517986$, $\pm0.929508517986$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{10})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $3$ |
| Isomorphism classes: | 6 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $90$ | $2340$ | $136890$ | $5475600$ | $280532250$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $50$ | $396$ | $2278$ | $16692$ | $117650$ | $825396$ | $5766718$ | $40336812$ | $282475250$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 3 curves (of which all are hyperelliptic):
- $y^2=3 x^6+4 x^5+3 x^4+6 x^3+4 x^2+3 x$
- $y^2=5 x^6+6 x^5+4 x^4+4 x^2+x+5$
- $y^2=3 x^5+3 x^4+3 x^3+2 x^2+6 x+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7^{4}}$.
Endomorphism algebra over $\F_{7}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{10})\). |
| The base change of $A$ to $\F_{7^{4}}$ is 1.2401.ack 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-10}) \)$)$ |
- Endomorphism algebra over $\F_{7^{2}}$
The base change of $A$ to $\F_{7^{2}}$ is the simple isogeny class 2.49.a_ack and its endomorphism algebra is \(\Q(i, \sqrt{10})\).
Base change
This is a primitive isogeny class.