Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 81 x^{2} + 2809 x^{4}$ |
| Frobenius angles: | $\pm0.111579124397$, $\pm0.888420875603$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{187})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $18$ |
| Isomorphism classes: | 28 |
| Cyclic group of points: | yes |
This isogeny class is simple but not geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2729$ | $7447441$ | $22164512276$ | $62244825632521$ | $174887471147172689$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $54$ | $2648$ | $148878$ | $7888596$ | $418195494$ | $22164663422$ | $1174711139838$ | $62259720194788$ | $3299763591802134$ | $174887471928832328$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):
- $y^2=10 x^6+45 x^5+38 x^4+51 x^3+8 x^2+34 x+5$
- $y^2=22 x^6+17 x^5+13 x^4+34 x^3+48 x^2+16 x+40$
- $y^2=6 x^6+13 x^5+29 x^4+5 x^3+33 x^2+37 x+2$
- $y^2=17 x^6+40 x^5+31 x^4+10 x^3+30 x^2+45 x+41$
- $y^2=34 x^6+27 x^5+9 x^4+20 x^3+7 x^2+37 x+29$
- $y^2=20 x^6+46 x^5+46 x^4+25 x^3+6 x^2+25 x+11$
- $y^2=40 x^6+39 x^5+39 x^4+50 x^3+12 x^2+50 x+22$
- $y^2=22 x^6+23 x^5+28 x^4+30 x^3+33 x^2+33 x+17$
- $y^2=44 x^6+46 x^5+3 x^4+7 x^3+13 x^2+13 x+34$
- $y^2=9 x^6+16 x^5+43 x^4+13 x^3+8 x^2+6 x+45$
- $y^2=13 x^6+27 x^5+23 x^4+26 x^3+46 x^2+23 x+8$
- $y^2=26 x^6+x^5+46 x^4+52 x^3+39 x^2+46 x+16$
- $y^2=15 x^6+47 x^5+38 x^4+13 x^3+5 x^2+17 x+23$
- $y^2=31 x^6+16 x^5+8 x^4+11 x^3+24 x^2+38 x+42$
- $y^2=16 x^6+29 x^5+44 x^4+30 x^2+5 x+45$
- $y^2=32 x^6+5 x^5+35 x^4+7 x^2+10 x+37$
- $y^2=16 x^6+44 x^5+9 x^4+10 x^3+29 x^2+37 x+18$
- $y^2=32 x^6+35 x^5+18 x^4+20 x^3+5 x^2+21 x+36$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{53^{2}}$.
Endomorphism algebra over $\F_{53}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{187})\). |
| The base change of $A$ to $\F_{53^{2}}$ is 1.2809.add 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-187}) \)$)$ |
Base change
This is a primitive isogeny class.