Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 7 x + 53 x^{2} )^{2}$ |
| $1 - 14 x + 155 x^{2} - 742 x^{3} + 2809 x^{4}$ | |
| Frobenius angles: | $\pm0.340360113580$, $\pm0.340360113580$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $7$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2209$ | $8219689$ | $22394523904$ | $62297096908201$ | $174867208544385289$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $40$ | $2924$ | $150418$ | $7895220$ | $418147040$ | $22163770838$ | $1174709575856$ | $62259710748964$ | $3299763817056394$ | $174887470864399964$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 7 curves (of which all are hyperelliptic):
- $y^2=32 x^6+39 x^5+9 x^4+15 x^3+40 x^2+50 x+50$
- $y^2=7 x^6+32 x^5+3 x^4+46 x^3+39 x^2+2 x+9$
- $y^2=50 x^6+42 x^5+24 x^4+45 x^3+52 x^2+19 x+49$
- $y^2=33 x^6+10 x^5+50 x^4+38 x^3+28 x^2+x+5$
- $y^2=7 x^6+12 x^5+43 x^4+51 x^3+52 x^2+31 x+23$
- $y^2=26 x^6+48 x^5+42 x^4+21 x^3+31 x^2+39 x+3$
- $y^2=22 x^6+52 x^5+36 x^4+13 x^3+5 x^2+28 x+51$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{53}$.
Endomorphism algebra over $\F_{53}$| The isogeny class factors as 1.53.ah 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-163}) \)$)$ |
Base change
This is a primitive isogeny class.