Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 7 x - 10 x^{2} + 413 x^{3} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.317263801822$, $\pm0.983930468489$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-187})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3892$ | $11878384$ | $42549788176$ | $146803760017600$ | $511143532908249052$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $67$ | $3413$ | $207172$ | $12115161$ | $714961757$ | $42179749526$ | $2488652019263$ | $146830418214001$ | $8662996153183708$ | $511116753273819653$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=26 x^6+6 x^5+12 x^4+53 x^3+50 x^2+57 x+50$
- $y^2=8 x^6+51 x^5+31 x^4+43 x^3+32 x^2+7 x+26$
- $y^2=52 x^6+30 x^5+17 x^4+10 x^3+58 x^2+58 x+22$
- $y^2=14 x^6+26 x^5+17 x^4+31 x^3+12 x^2+20 x+17$
- $y^2=47 x^6+5 x^5+36 x^4+52 x^3+6 x^2+46 x+39$
- $y^2=12 x^6+4 x^5+53 x^4+52 x^3+47 x^2+27 x+21$
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 \(\Q(\sqrt{-3}, \sqrt{-187})\). |
| The base change of $A$ to $\F_{59^{3}}$ is 1.205379.bim 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-187}) \)$)$ |
Base change
This is a primitive isogeny class.