Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 24 x + 254 x^{2} - 1416 x^{3} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.0841624533377$, $\pm0.296351869071$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.39168.3 |
| Galois group: | $D_{4}$ |
| Jacobians: | $28$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2296$ | $11884096$ | $42224764792$ | $146859475166208$ | $511109531095036216$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $36$ | $3414$ | $205596$ | $12119758$ | $714914196$ | $42180225126$ | $2488649287692$ | $146830440588574$ | $8662996046660868$ | $511116755977594614$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):
- $y^2=31 x^6+45 x^5+27 x^4+46 x^3+15 x^2+25 x+40$
- $y^2=6 x^6+8 x^5+42 x^4+48 x^3+12 x^2+40 x$
- $y^2=38 x^6+21 x^5+17 x^4+15 x^3+4 x^2+57 x+34$
- $y^2=37 x^6+25 x^5+2 x^4+58 x^3+38 x^2+40$
- $y^2=47 x^6+6 x^5+33 x^4+18 x^2+8 x+1$
- $y^2=22 x^6+7 x^5+31 x^4+14 x^3+20 x^2+4 x+30$
- $y^2=6 x^6+9 x^5+55 x^4+25 x^3+24 x^2+33 x+43$
- $y^2=11 x^6+45 x^5+54 x^4+48 x^3+20 x^2+20 x+12$
- $y^2=x^6+x^5+49 x^4+x^3+40 x^2+31 x+57$
- $y^2=11 x^6+34 x^5+5 x^4+52 x^3+31 x^2+9 x+58$
- $y^2=21 x^6+9 x^5+18 x^4+7 x^3+9 x^2+x+31$
- $y^2=11 x^6+5 x^5+39 x^4+4 x^3+45 x^2+48 x+56$
- $y^2=52 x^6+15 x^5+9 x^4+58 x^3+47 x^2+46 x+2$
- $y^2=39 x^6+40 x^5+16 x^4+7 x^3+49 x^2+21 x+36$
- $y^2=55 x^6+41 x^5+17 x^4+38 x^3+22 x^2+2 x+25$
- $y^2=29 x^6+33 x^5+22 x^4+52 x^3+45 x^2+9 x+29$
- $y^2=17 x^6+x^5+51 x^4+16 x^3+52 x^2+55 x+45$
- $y^2=23 x^6+55 x^5+24 x^4+3 x^3+29 x^2+47 x+40$
- $y^2=6 x^6+11 x^5+51 x^4+40 x^3+11 x^2+12 x+1$
- $y^2=2 x^6+28 x^5+43 x^4+9 x^3+21 x^2+39 x+3$
- $y^2=39 x^6+7 x^5+51 x^4+50 x^3+45 x^2+54 x+24$
- $y^2=44 x^6+49 x^5+3 x^4+25 x^3+36 x^2+26 x+10$
- $y^2=52 x^6+11 x^5+13 x^4+36 x^3+16 x^2+6 x+13$
- $y^2=34 x^6+12 x^5+22 x^4+42 x^3+44 x^2+37 x+2$
- $y^2=23 x^6+10 x^5+20 x^4+22 x^3+53 x^2+54 x+48$
- $y^2=20 x^6+20 x^5+18 x^4+51 x^3+2 x^2+49 x+54$
- $y^2=56 x^6+17 x^5+x^4+18 x^3+43 x^2+32 x+11$
- $y^2=10 x^6+51 x^5+20 x^4+10 x^3+2 x^2+9 x+56$
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 4.0.39168.3. |
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_ju | $2$ | (not in LMFDB) |