Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 10 x + 59 x^{2} )^{2}$ |
| $1 + 20 x + 218 x^{2} + 1180 x^{3} + 3481 x^{4}$ | |
| Frobenius angles: | $\pm0.725626973200$, $\pm0.725626973200$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $26$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 5, 7$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4900$ | $12250000$ | $41865252100$ | $146991376000000$ | $511086799841222500$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $80$ | $3518$ | $203840$ | $12130638$ | $714882400$ | $42180169358$ | $2488657599760$ | $146830397947678$ | $8662995854439920$ | $511116755282533598$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 26 curves (of which all are hyperelliptic):
- $y^2=32 x^6+12 x^5+55 x^4+13 x^3+58 x^2+45 x+30$
- $y^2=6 x^6+56 x^5+44 x^4+56 x^3+58 x^2+11 x+30$
- $y^2=55 x^6+51 x^5+38 x^4+28 x^3+43 x^2+28 x+50$
- $y^2=22 x^6+36 x^5+55 x^4+11 x^3+46 x^2+2 x+15$
- $y^2=29 x^6+10 x^5+18 x^4+41 x^3+25 x^2+12 x+11$
- $y^2=13 x^6+58 x^5+33 x^4+34 x^3+56 x^2+8 x+20$
- $y^2=21 x^6+29 x^5+29 x^4+40 x^3+19 x^2+32 x+16$
- $y^2=15 x^6+6 x^5+18 x^4+51 x^3+18 x^2+6 x+15$
- $y^2=47 x^6+2 x^5+56 x^4+21 x^3+14 x^2+37 x+40$
- $y^2=50 x^5+57 x^4+55 x^3+28 x^2+48 x+52$
- $y^2=56 x^5+27 x^4+18 x^3+27 x^2+56 x$
- $y^2=9 x^6+30 x^4+30 x^2+9$
- $y^2=34 x^6+21 x^5+46 x^4+51 x^3+46 x^2+21 x+34$
- $y^2=16 x^6+33 x^5+2 x^4+57 x^3+42 x^2+39 x+27$
- $y^2=5 x^6+35 x^5+x^4+7 x^3+15 x^2+53 x+28$
- $y^2=45 x^6+27 x^5+25 x^4+9 x^3+25 x^2+27 x+45$
- $y^2=18 x^6+31 x^5+32 x^4+52 x^3+9 x^2+53 x+27$
- $y^2=14 x^6+25 x^5+31 x^4+20 x^3+37 x^2+37 x+26$
- $y^2=54 x^6+24 x^5+34 x^4+26 x^3+12 x^2+17 x+39$
- $y^2=19 x^6+31 x^5+40 x^4+51 x^3+40 x^2+31 x+19$
- $y^2=7 x^6+56 x^5+55 x^4+51 x^3+51 x^2+54 x+25$
- $y^2=51 x^5+53 x^4+4 x^3+45 x^2+54 x+1$
- $y^2=18 x^6+9 x^5+30 x^4+45 x^3+30 x^2+9 x+18$
- $y^2=55 x^6+12 x^5+15 x^4+x^3+46 x^2+57 x+44$
- $y^2=22 x^6+7 x^5+19 x^4+24 x^3+19 x^2+7 x+22$
- $y^2=56 x^6+23 x^4+23 x^2+56$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{59}$.
Endomorphism algebra over $\F_{59}$| The isogeny class factors as 1.59.k 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-34}) \)$)$ |
Base change
This is a primitive isogeny class.