Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + x + 53 x^{2} )^{2}$ |
| $1 + 2 x + 107 x^{2} + 106 x^{3} + 2809 x^{4}$ | |
| Frobenius angles: | $\pm0.521878836125$, $\pm0.521878836125$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $30$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $5, 11$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3025$ | $8497225$ | $22117638400$ | $62174407755625$ | $174898997696025625$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $56$ | $3020$ | $148562$ | $7879668$ | $418223056$ | $22164906710$ | $1174709133472$ | $62259663501988$ | $3299763725048906$ | $174887471658463100$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 30 curves (of which all are hyperelliptic):
- $y^2=28 x^6+50 x^5+11 x^4+16 x^3+28 x^2+17 x+12$
- $y^2=18 x^6+49 x^5+3 x^4+52 x^3+48 x^2+36 x+5$
- $y^2=51 x^6+18 x^5+13 x^4+42 x^3+17 x^2+22 x+19$
- $y^2=45 x^6+50 x^5+20 x^4+12 x^3+18 x^2+14 x+23$
- $y^2=33 x^6+2 x^5+x^4+37 x^3+32 x^2+43 x+38$
- $y^2=16 x^6+27 x^5+38 x^4+33 x^3+41 x^2+21 x+43$
- $y^2=28 x^6+46 x^5+6 x^4+37 x^3+35 x^2+12 x+30$
- $y^2=11 x^6+21 x^5+45 x^4+46 x^3+35 x^2+12 x+9$
- $y^2=8 x^6+50 x^5+49 x^4+25 x^3+36 x^2+22 x+51$
- $y^2=19 x^6+5 x^5+4 x^4+34 x^3+36 x^2+3 x+14$
- $y^2=25 x^6+15 x^4+20 x^3+38 x^2+43 x+6$
- $y^2=48 x^6+44 x^5+10 x^4+27 x^2+22 x+41$
- $y^2=9 x^6+50 x^5+12 x^4+4 x^3+9 x^2+18 x+27$
- $y^2=43 x^6+14 x^5+33 x^4+37 x^3+27 x^2+12 x+4$
- $y^2=27 x^6+17 x^5+3 x^4+45 x^3+45 x^2+9 x+18$
- $y^2=48 x^6+30 x^5+47 x^4+11 x^3+47 x^2+30 x+48$
- $y^2=7 x^6+44 x^5+45 x^4+38 x^3+35 x^2+24 x+11$
- $y^2=21 x^6+14 x^5+10 x^4+50 x^3+42 x^2+34 x+38$
- $y^2=16 x^6+32 x^5+27 x^4+24 x^3+30 x^2+42 x+27$
- $y^2=51 x^6+9 x^5+47 x^4+36 x^3+21 x^2+33 x+11$
- $y^2=42 x^6+6 x^5+44 x^4+23 x^3+2 x^2+19 x+25$
- $y^2=22 x^6+18 x^5+9 x^4+x^3+x^2+12 x+35$
- $y^2=20 x^6+35 x^5+33 x^4+33 x^3+2 x^2+3 x+18$
- $y^2=38 x^6+51 x^5+12 x^4+31 x^3+41 x^2+51 x+15$
- $y^2=2 x^6+40 x^5+33 x^4+33 x^3+12 x^2+37 x+3$
- $y^2=19 x^6+42 x^5+42 x^4+11 x^3+32 x^2+30 x+22$
- $y^2=14 x^6+10 x^5+48 x^4+37 x^3+2 x^2+44 x+22$
- $y^2=42 x^6+23 x^5+39 x^4+42 x^3+48 x^2+30 x+25$
- $y^2=3 x^6+39 x^5+10 x^4+40 x^3+30 x^2+39 x+3$
- $y^2=11 x^6+30 x^5+43 x^4+33 x^3+12 x^2+6 x+40$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{53}$.
Endomorphism algebra over $\F_{53}$| The isogeny class factors as 1.53.b 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-211}) \)$)$ |
Base change
This is a primitive isogeny class.