Invariants
| Base field: | $\F_{53}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 14 x + 53 x^{2} )( 1 - 7 x + 53 x^{2} )$ |
| $1 - 21 x + 204 x^{2} - 1113 x^{3} + 2809 x^{4}$ | |
| Frobenius angles: | $\pm0.0885855327829$, $\pm0.340360113580$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $6$ |
| Isomorphism classes: | 38 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $1880$ | $7798240$ | $22201777280$ | $62258808688000$ | $174874289438449400$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $33$ | $2777$ | $149130$ | $7890369$ | $418163973$ | $22164095414$ | $1174711156449$ | $62259710200801$ | $3299763796792290$ | $174887471398145057$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=12x^6+49x^5+14x^4+28x^3+23x^2+44x+30$
- $y^2=33x^6+5x^4+23x^3+33x^2+12x+26$
- $y^2=31x^6+48x^5+14x^4+23x^3+9x^2+33x+51$
- $y^2=35x^6+6x^5+50x^4+38x^3+40x^2+30x+45$
- $y^2=x^6+28x^5+40x^4+33x^3+x^2+15x+21$
- $y^2=33x^6+4x^5+39x^4+50x^3+31x^2+27x+42$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{53}$.
Endomorphism algebra over $\F_{53}$| The isogeny class factors as 1.53.ao $\times$ 1.53.ah and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.