Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 8 x + 5 x^{2} - 472 x^{3} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.00768306809500$, $\pm0.658983598572$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-43})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $8$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3007$ | $11928769$ | $41810434576$ | $146781415010425$ | $511101762492651727$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $52$ | $3428$ | $203572$ | $12113316$ | $714903332$ | $42179720726$ | $2488649470268$ | $146830429739716$ | $8662995455104108$ | $511116752310449828$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=7 x^6+34 x^5+50 x^4+3 x^3+45 x^2+10 x+3$
- $y^2=51 x^6+51 x^5+40 x^4+37 x^3+29 x^2+35 x+48$
- $y^2=47 x^6+x^5+45 x^4+43 x^3+54 x^2+6 x+36$
- $y^2=56 x^6+58 x^5+29 x^4+37 x^3+58 x^2+4 x+57$
- $y^2=45 x^6+22 x^5+28 x^4+54 x^3+37 x^2+43 x+3$
- $y^2=17 x^6+55 x^4+42 x^3+40 x^2+34 x+43$
- $y^2=30 x^6+48 x^5+14 x^4+44 x^3+57 x^2+44 x+41$
- $y^2=26 x^6+13 x^5+39 x^4+5 x^3+2 x^2+34 x+41$
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 \(\Q(\sqrt{-3}, \sqrt{-43})\). |
| The base change of $A$ to $\F_{59^{3}}$ is 1.205379.abiu 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$ |
Base change
This is a primitive isogeny class.