Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 6 x - 23 x^{2} + 354 x^{3} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.294387598742$, $\pm0.961054265409$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $7$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3819$ | $11835081$ | $42529163076$ | $146827541611449$ | $511151418756219099$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $66$ | $3400$ | $207072$ | $12117124$ | $714972786$ | $42179923726$ | $2488650453894$ | $146830413426244$ | $8662995987142608$ | $511116754221685000$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which all are hyperelliptic):
- $y^2=41 x^6+47 x^5+36 x^4+11 x^3+3 x^2+22 x+41$
- $y^2=4 x^6+40 x^5+13 x^4+43 x^3+50 x^2+43 x+4$
- $y^2=42 x^6+52 x^5+47 x^4+18 x^3+4 x^2+23 x+42$
- $y^2=33 x^6+16 x^5+49 x^4+53 x^3+51 x^2+5 x+33$
- $y^2=2 x^6+28 x^5+53 x^4+29 x^3+46 x^2+19 x+2$
- $y^2=9 x^6+34 x^5+25 x^4+29 x^3+49 x^2+20 x+9$
- $y^2=49 x^6+11 x^5+3 x^4+28 x^3+34 x^2+47 x+49$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{59^{3}}$.
Endomorphism algebra over $\F_{59}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-2}, \sqrt{-3})\). |
| The base change of $A$ to $\F_{59^{3}}$ is 1.205379.bgo 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |
Base change
This is a primitive isogeny class.