Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 5 x + 59 x^{2} )^{2}$ |
| $1 + 10 x + 143 x^{2} + 590 x^{3} + 3481 x^{4}$ | |
| Frobenius angles: | $\pm0.605523279018$, $\pm0.605523279018$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $30$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $5, 13$ |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4225$ | $12780625$ | $41869344400$ | $146789580705625$ | $511192932752880625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $70$ | $3668$ | $203860$ | $12113988$ | $715030850$ | $42180199958$ | $2488646866790$ | $146830480381828$ | $8662995877231180$ | $511116750483887348$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):
- $y^2=32 x^6+12 x^5+25 x^4+29 x^3+32 x^2+3 x+53$
- $y^2=33 x^6+9 x^5+31 x^4+43 x^3+43 x^2+3 x+35$
- $y^2=33 x^6+48 x^5+40 x^4+21 x^3+40 x^2+45 x+58$
- $y^2=26 x^6+40 x^5+4 x^4+5 x^3+39 x^2+24 x+22$
- $y^2=51 x^6+54 x^5+37 x^4+18 x^3+51 x^2+45 x+1$
- $y^2=41 x^6+17 x^5+18 x^4+45 x^3+24 x^2+42 x+3$
- $y^2=52 x^6+22 x^5+42 x^4+53 x^3+42 x^2+22 x+52$
- $y^2=6 x^6+28 x^5+39 x^4+38 x^3+39 x^2+28 x+6$
- $y^2=22 x^6+50 x^5+29 x^4+43 x^3+29 x^2+47 x+7$
- $y^2=21 x^6+38 x^5+21 x^4+24 x^3+21 x^2+38 x+21$
- $y^2=43 x^6+52 x^5+21 x^4+25 x^3+56 x^2+46 x+34$
- $y^2=21 x^6+55 x^5+27 x^4+2 x^3+57 x^2+40 x+3$
- $y^2=34 x^6+43 x^5+46 x^4+21 x^3+35 x^2+52 x+56$
- $y^2=4 x^6+42 x^5+14 x^4+10 x^3+8 x^2+32 x+16$
- $y^2=45 x^6+35 x^5+29 x^3+35 x+45$
- $y^2=49 x^6+48 x^5+5 x^4+4 x^3+17 x^2+20 x+25$
- $y^2=47 x^6+21 x^5+46 x^4+36 x^3+49 x^2+5 x+7$
- $y^2=41 x^6+47 x^5+21 x^4+16 x^3+21 x^2+47 x+41$
- $y^2=10 x^6+43 x^5+58 x^4+35 x^3+37 x^2+27 x+42$
- $y^2=17 x^6+47 x^5+46 x^4+9 x^3+10 x^2+2 x+14$
- $y^2=21 x^6+11 x^5+48 x^4+58 x^3+48 x^2+11 x+21$
- $y^2=35 x^6+10 x^5+7 x^4+13 x^3+7 x^2+10 x+35$
- $y^2=12 x^6+47 x^5+33 x^4+8 x^3+33 x^2+47 x+12$
- $y^2=13 x^6+25 x^5+4 x^4+x^3+40 x^2+41$
- $y^2=24 x^6+40 x^5+20 x^4+30 x^3+20 x^2+40 x+24$
- $y^2=42 x^6+42 x^5+41 x^4+16 x^3+41 x^2+42 x+42$
- $y^2=49 x^6+24 x^5+44 x^4+34 x^3+44 x^2+24 x+49$
- $y^2=28 x^6+52 x^5+30 x^4+9 x^3+34 x^2+2 x+31$
- $y^2=51 x^6+50 x^5+47 x^4+24 x^3+47 x^2+50 x+51$
- $y^2=16 x^6+54 x^5+22 x^4+58 x^3+28 x^2+41 x+31$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{59}$.
Endomorphism algebra over $\F_{59}$| The isogeny class factors as 1.59.f 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-211}) \)$)$ |
Base change
This is a primitive isogeny class.