Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 23 x + 249 x^{2} - 1357 x^{3} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.193213474443$, $\pm0.263794663845$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.246125.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Cyclic group of points: | yes |
This isogeny class is simple and 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2351$ | $12015961$ | $42374779001$ | $146977372478045$ | $511175360036549296$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $37$ | $3451$ | $206323$ | $12129483$ | $715006272$ | $42180788431$ | $2488650099553$ | $146830411343523$ | $8662995619992187$ | $511116752503609206$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=40 x^6+36 x^5+34 x^4+18 x^3+3 x^2+34 x+50$
- $y^2=15 x^6+25 x^5+25 x^4+24 x^3+25 x^2+24 x+37$
- $y^2=39 x^6+30 x^5+15 x^4+51 x^3+18 x^2+12 x+12$
- $y^2=8 x^6+56 x^5+4 x^4+47 x^3+2 x^2+9 x+44$
- $y^2=52 x^6+26 x^5+58 x^4+55 x^3+33 x^2+21 x+2$
- $y^2=32 x^6+50 x^5+41 x^4+50 x^3+45 x^2+13 x+10$
- $y^2=10 x^6+27 x^5+55 x^4+12 x^3+43 x^2+53 x+33$
- $y^2=44 x^6+15 x^5+25 x^4+15 x^3+40 x^2+44 x+25$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{59}$.
Endomorphism algebra over $\F_{59}$| The endomorphism algebra of this simple isogeny class is 4.0.246125.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.59.x_jp | $2$ | (not in LMFDB) |