Invariants
| Base field: | $\F_{59}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 8 x + 59 x^{2} )^{2}$ |
| $1 - 16 x + 182 x^{2} - 944 x^{3} + 3481 x^{4}$ | |
| Frobenius angles: | $\pm0.325650265238$, $\pm0.325650265238$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $30$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2704$ | $12503296$ | $42553088656$ | $146928531902464$ | $511086774104702224$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $44$ | $3590$ | $207188$ | $12125454$ | $714882364$ | $42179720726$ | $2488647455716$ | $146830453333534$ | $8662996182205772$ | $511116755281024550$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):
- $y^2=55 x^6+7 x^5+3 x^4+29 x^3+20$
- $y^2=43 x^6+29 x^5+x^4+4 x^3+49 x^2+51 x+46$
- $y^2=16 x^6+12 x^5+38 x^4+5 x^3+13 x^2+20 x+25$
- $y^2=20 x^6+x^5+55 x^4+50 x^3+55 x^2+x+20$
- $y^2=x^6+37 x^5+26 x^4+7 x^3+26 x^2+37 x+1$
- $y^2=50 x^6+56 x^5+20 x^4+8 x^3+20 x^2+56 x+50$
- $y^2=21 x^6+36 x^5+33 x^4+23 x^3+48 x^2+28 x+24$
- $y^2=27 x^6+51 x^5+38 x^4+56 x^3+50 x^2+41 x+20$
- $y^2=57 x^6+18 x^4+18 x^2+57$
- $y^2=42 x^6+18 x^5+58 x^4+5 x^3+21 x^2+3 x+28$
- $y^2=11 x^6+12 x^4+12 x^2+11$
- $y^2=3 x^6+36 x^5+16 x^4+49 x^3+45 x^2+10 x+2$
- $y^2=58 x^6+23 x^5+37 x^4+33 x^3+42 x^2+29 x+9$
- $y^2=58 x^6+20 x^5+43 x^4+44 x^3+3 x^2+57 x+10$
- $y^2=12 x^6+56 x^5+48 x^4+24 x^3+17 x^2+24 x+22$
- $y^2=6 x^6+46 x^5+24 x^4+31 x^3+13 x^2+x+44$
- $y^2=38 x^6+23 x^5+8 x^4+35 x^3+30 x^2+27 x+16$
- $y^2=13 x^6+46 x^5+30 x^4+29 x^3+30 x^2+46 x+13$
- $y^2=43 x^6+7 x^5+58 x^4+34 x^3+58 x^2+7 x+43$
- $y^2=15 x^6+30 x^5+35 x^4+8 x^3+57 x^2+42 x+46$
- $y^2=7 x^6+56 x^5+4 x^4+31 x^3+34 x^2+26 x+25$
- $y^2=14 x^6+47 x^5+6 x^4+12 x^3+16 x^2+32 x+2$
- $y^2=35 x^6+10 x^4+10 x^2+35$
- $y^2=28 x^6+47 x^5+3 x^4+18 x^3+18 x^2+44 x+6$
- $y^2=7 x^6+58 x^5+22 x^4+55 x^3+14 x^2+2 x+30$
- $y^2=42 x^6+32 x^5+54 x^3+33 x^2+17 x+7$
- $y^2=51 x^5+21 x^4+36 x^3+53 x^2+5 x+18$
- $y^2=8 x^6+5 x^5+5 x^4+57 x^3+48 x^2+57 x+22$
- $y^2=9 x^6+8 x^4+54 x^3+58 x^2+33 x+54$
- $y^2=20 x^6+54 x^5+55 x^4+39 x^3+41 x^2+18 x+43$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{59}$.
Endomorphism algebra over $\F_{59}$| The isogeny class factors as 1.59.ai 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$ |
Base change
This is a primitive isogeny class.