Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 24 x + 257 x^{2} - 1416 x^{3} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.122641944864$, $\pm0.280761917494$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.117225.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $20$ |
| Isomorphism classes: | 20 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2299$ | $11906521$ | $42269377216$ | $146906097082569$ | $511141445652468499$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $36$ | $3420$ | $205812$ | $12123604$ | $714958836$ | $42180585846$ | $2488651196844$ | $146830444451044$ | $8662996010687148$ | $511116755560310700$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):
- $y^2=56 x^6+37 x^5+13 x^4+32 x^3+16 x^2+40 x+48$
- $y^2=38 x^6+45 x^5+30 x^4+42 x^3+56 x^2+27 x+6$
- $y^2=26 x^6+28 x^5+34 x^4+30 x^3+34 x^2+43 x+30$
- $y^2=28 x^6+5 x^5+3 x^4+7 x^3+3 x^2+18 x+33$
- $y^2=42 x^6+9 x^5+51 x^4+8 x^3+18 x+34$
- $y^2=8 x^6+47 x^5+15 x^4+15 x^3+15 x^2+4 x+8$
- $y^2=22 x^6+25 x^5+45 x^4+33 x^3+11 x+33$
- $y^2=3 x^6+6 x^4+32 x^3+3 x^2+19 x+30$
- $y^2=34 x^6+36 x^5+21 x^4+51 x^3+27 x^2+23 x+23$
- $y^2=27 x^6+27 x^5+44 x^4+23 x^3+2 x^2+39 x+41$
- $y^2=8 x^6+32 x^5+35 x^4+16 x^3+33 x^2+12 x+16$
- $y^2=33 x^6+8 x^5+33 x^4+18 x^3+8 x^2+12 x+2$
- $y^2=6 x^6+19 x^5+14 x^4+39 x^3+45 x+26$
- $y^2=32 x^6+30 x^5+22 x^4+x^3+15 x^2+8 x+51$
- $y^2=34 x^6+35 x^5+54 x^4+40 x^3+5 x^2+12 x+15$
- $y^2=43 x^6+52 x^5+28 x^4+55 x^3+29 x^2+51 x+55$
- $y^2=21 x^6+2 x^5+55 x^4+17 x^3+53 x^2+20 x+30$
- $y^2=58 x^6+40 x^5+4 x^4+6 x^3+10 x^2+13$
- $y^2=43 x^6+4 x^5+47 x^4+5 x^3+46 x^2+32 x+55$
- $y^2=43 x^6+47 x^5+12 x^4+48 x^3+9 x^2+10 x+32$
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 4.0.117225.1. |
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.y_jx | $2$ | (not in LMFDB) |