Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 24 x + 254 x^{2} + 1416 x^{3} + 3481 x^{4}$ |
| Frobenius angles: | $\pm0.703648130929$, $\pm0.915837546662$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.39168.3 |
| Galois group: | $D_{4}$ |
| Jacobians: | $28$ |
| Isomorphism classes: | 44 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $5176$ | $11884096$ | $42136040632$ | $146859475166208$ | $511123978285290616$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $84$ | $3414$ | $205164$ | $12119758$ | $714934404$ | $42180225126$ | $2488653681948$ | $146830440588574$ | $8662995590649012$ | $511116755977594614$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 28 curves (of which all are hyperelliptic):
- $y^2=3 x^6+31 x^5+54 x^4+33 x^3+30 x^2+50 x+21$
- $y^2=12 x^6+16 x^5+25 x^4+37 x^3+24 x^2+21 x$
- $y^2=17 x^6+42 x^5+34 x^4+30 x^3+8 x^2+55 x+9$
- $y^2=48 x^6+42 x^5+x^4+29 x^3+19 x^2+20$
- $y^2=53 x^6+3 x^5+46 x^4+9 x^2+4 x+30$
- $y^2=11 x^6+33 x^5+45 x^4+7 x^3+10 x^2+2 x+15$
- $y^2=3 x^6+34 x^5+57 x^4+42 x^3+12 x^2+46 x+51$
- $y^2=22 x^6+31 x^5+49 x^4+37 x^3+40 x^2+40 x+24$
- $y^2=2 x^6+2 x^5+39 x^4+2 x^3+21 x^2+3 x+55$
- $y^2=35 x^6+17 x^5+32 x^4+26 x^3+45 x^2+34 x+29$
- $y^2=40 x^6+34 x^5+9 x^4+33 x^3+34 x^2+30 x+45$
- $y^2=35 x^6+32 x^5+49 x^4+2 x^3+52 x^2+24 x+28$
- $y^2=26 x^6+37 x^5+34 x^4+29 x^3+53 x^2+23 x+1$
- $y^2=19 x^6+21 x^5+32 x^4+14 x^3+39 x^2+42 x+13$
- $y^2=51 x^6+23 x^5+34 x^4+17 x^3+44 x^2+4 x+50$
- $y^2=44 x^6+46 x^5+11 x^4+26 x^3+52 x^2+34 x+44$
- $y^2=34 x^6+2 x^5+43 x^4+32 x^3+45 x^2+51 x+31$
- $y^2=41 x^6+57 x^5+12 x^4+31 x^3+44 x^2+53 x+20$
- $y^2=12 x^6+22 x^5+43 x^4+21 x^3+22 x^2+24 x+2$
- $y^2=x^6+14 x^5+51 x^4+34 x^3+40 x^2+49 x+31$
- $y^2=19 x^6+14 x^5+43 x^4+41 x^3+31 x^2+49 x+48$
- $y^2=29 x^6+39 x^5+6 x^4+50 x^3+13 x^2+52 x+20$
- $y^2=26 x^6+35 x^5+36 x^4+18 x^3+8 x^2+3 x+36$
- $y^2=17 x^6+6 x^5+11 x^4+21 x^3+22 x^2+48 x+1$
- $y^2=46 x^6+20 x^5+40 x^4+44 x^3+47 x^2+49 x+37$
- $y^2=40 x^6+40 x^5+36 x^4+43 x^3+4 x^2+39 x+49$
- $y^2=53 x^6+34 x^5+2 x^4+36 x^3+27 x^2+5 x+22$
- $y^2=20 x^6+43 x^5+40 x^4+20 x^3+4 x^2+18 x+53$
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.ay_ju | $2$ | (not in LMFDB) |