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: | $\pm0.674349734762$, $\pm0.674349734762$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $30$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 17$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $4624$ | $12503296$ | $41810434576$ | $146928531902464$ | $511146736235591824$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $76$ | $3590$ | $203572$ | $12125454$ | $714966236$ | $42179720726$ | $2488655513924$ | $146830453333534$ | $8662995455104108$ | $511116755281024550$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):
- $y^2=57 x^6+33 x^5+31 x^4+44 x^3+10$
- $y^2=27 x^6+58 x^5+2 x^4+8 x^3+39 x^2+43 x+33$
- $y^2=32 x^6+24 x^5+17 x^4+10 x^3+26 x^2+40 x+50$
- $y^2=40 x^6+2 x^5+51 x^4+41 x^3+51 x^2+2 x+40$
- $y^2=2 x^6+15 x^5+52 x^4+14 x^3+52 x^2+15 x+2$
- $y^2=25 x^6+28 x^5+10 x^4+4 x^3+10 x^2+28 x+25$
- $y^2=40 x^6+18 x^5+46 x^4+41 x^3+24 x^2+14 x+12$
- $y^2=54 x^6+43 x^5+17 x^4+53 x^3+41 x^2+23 x+40$
- $y^2=55 x^6+36 x^4+36 x^2+55$
- $y^2=25 x^6+36 x^5+57 x^4+10 x^3+42 x^2+6 x+56$
- $y^2=22 x^6+24 x^4+24 x^2+22$
- $y^2=6 x^6+13 x^5+32 x^4+39 x^3+31 x^2+20 x+4$
- $y^2=57 x^6+46 x^5+15 x^4+7 x^3+25 x^2+58 x+18$
- $y^2=57 x^6+40 x^5+27 x^4+29 x^3+6 x^2+55 x+20$
- $y^2=24 x^6+53 x^5+37 x^4+48 x^3+34 x^2+48 x+44$
- $y^2=3 x^6+23 x^5+12 x^4+45 x^3+36 x^2+30 x+22$
- $y^2=17 x^6+46 x^5+16 x^4+11 x^3+x^2+54 x+32$
- $y^2=36 x^6+23 x^5+15 x^4+44 x^3+15 x^2+23 x+36$
- $y^2=51 x^6+33 x^5+29 x^4+17 x^3+29 x^2+33 x+51$
- $y^2=37 x^6+15 x^5+47 x^4+4 x^3+58 x^2+21 x+23$
- $y^2=14 x^6+53 x^5+8 x^4+3 x^3+9 x^2+52 x+50$
- $y^2=7 x^6+53 x^5+3 x^4+6 x^3+8 x^2+16 x+1$
- $y^2=11 x^6+20 x^4+20 x^2+11$
- $y^2=56 x^6+35 x^5+6 x^4+36 x^3+36 x^2+29 x+12$
- $y^2=33 x^6+29 x^5+11 x^4+57 x^3+7 x^2+x+15$
- $y^2=21 x^6+16 x^5+27 x^3+46 x^2+38 x+33$
- $y^2=43 x^5+42 x^4+13 x^3+47 x^2+10 x+36$
- $y^2=4 x^6+32 x^5+32 x^4+58 x^3+24 x^2+58 x+11$
- $y^2=34 x^6+4 x^4+27 x^3+29 x^2+46 x+27$
- $y^2=40 x^6+49 x^5+51 x^4+19 x^3+23 x^2+36 x+27$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{59}$.
Endomorphism algebra over $\F_{59}$| The isogeny class factors as 1.59.i 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$ |
Base change
This is a primitive isogeny class.