Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 2 x + 2 x^{2} + 14 x^{3} + 49 x^{4}$ |
| Frobenius angles: | $\pm0.336118664261$, $\pm0.836118664261$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{13})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $8$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $68$ | $2448$ | $131444$ | $5992704$ | $276522068$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $50$ | $382$ | $2494$ | $16450$ | $117650$ | $821110$ | $5770174$ | $40365274$ | $282475250$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=x^6+5 x^4+x^2+4 x+1$
- $y^2=5 x^6+6 x^5+2 x^4+3 x^3+4 x^2+6 x+4$
- $y^2=4 x^6+x^5+4 x^4+3 x^3+3 x^2+2 x+1$
- $y^2=5 x^5+x^4+4 x^2+5 x+1$
- $y^2=6 x^6+2 x^5+4 x^4+3 x^3+x^2+6 x+1$
- $y^2=x^6+x^5+3 x^4+3 x^3+3 x^2+2 x+1$
- $y^2=2 x^6+2 x^5+3 x^4+4 x^3+6 x^2+4 x+6$
- $y^2=5 x^5+x^4+x^2+2 x$
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 \(\Q(i, \sqrt{13})\). |
| The base change of $A$ to $\F_{7^{4}}$ is 1.2401.bu 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-13}) \)$)$ |
- Endomorphism algebra over $\F_{7^{2}}$
The base change of $A$ to $\F_{7^{2}}$ is the simple isogeny class 2.49.a_bu and its endomorphism algebra is \(\Q(i, \sqrt{13})\).
Base change
This is a primitive isogeny class.