Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 24 x + 260 x^{2} - 1416 x^{3} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.162046720973$, $\pm0.258016605436$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.444672.5 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| Isomorphism classes: | 8 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2302$ | $11928964$ | $42314003518$ | $146952287209872$ | $511171816790261422$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $36$ | $3426$ | $206028$ | $12127414$ | $715001316$ | $42180881010$ | $2488651775436$ | $146830428327070$ | $8662995744221124$ | $511116753128639586$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=2 x^6+46 x^5+2 x^4+57 x^3+28 x+13$
- $y^2=12 x^6+54 x^5+36 x^4+22 x^3+45 x^2+48 x+30$
- $y^2=47 x^6+53 x^5+37 x^4+44 x^3+51 x^2+56 x+28$
- $y^2=33 x^6+19 x^5+30 x^4+12 x^3+50 x^2+27 x+4$
- $y^2=53 x^5+29 x^4+24 x^3+41 x^2+10 x+15$
- $y^2=11 x^6+48 x^5+41 x^4+48 x^3+24 x^2+7 x+20$
- $y^2=20 x^6+43 x^5+47 x^4+38 x^3+36 x^2+30 x+1$
- $y^2=54 x^6+5 x^5+45 x^4+50 x^3+23 x^2+18 x+24$
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 4.0.444672.5. |
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.y_ka | $2$ | (not in LMFDB) |