Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 11 x + 53 x^{2} )^{2}$ |
| $1 - 22 x + 227 x^{2} - 1166 x^{3} + 2809 x^{4}$ | |
| Frobenius angles: | $\pm0.227402221936$, $\pm0.227402221936$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $9$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1849$ | $7812025$ | $22289295616$ | $62344842015625$ | $174918560400071569$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $32$ | $2780$ | $149714$ | $7901268$ | $418269832$ | $22164607190$ | $1174709906584$ | $62259663804388$ | $3299763364487882$ | $174887469275225900$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 9 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^6+24x^5+2x^4+16x^3+50x^2+x+33$
- $y^2=30x^6+28x^5+39x^4+48x^3+2x^2+46x+20$
- $y^2=48x^6+14x^4+34x^3+19x^2+18$
- $y^2=23x^6+49x^5+51x^4+11x^3+28x^2+47x+15$
- $y^2=37x^6+33x^5+8x^4+46x^3+39x^2+33x+18$
- $y^2=39x^6+40x^5+8x^4+10x^3+2x^2+17x+20$
- $y^2=28x^6+14x^5+x^4+2x^3+40x^2+34x+17$
- $y^2=33x^6+8x^5+51x^4+15x^3+15x^2+33x+43$
- $y^2=14x^6+48x^5+21x^4+10x^3+26x^2+35x+20$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{53}$.
Endomorphism algebra over $\F_{53}$| The isogeny class factors as 1.53.al 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-91}) \)$)$ |
Base change
This is a primitive isogeny class.