Invariants
Base field: | $\F_{53}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 14 x + 53 x^{2} )( 1 - 9 x + 53 x^{2} )$ |
$1 - 23 x + 232 x^{2} - 1219 x^{3} + 2809 x^{4}$ | |
Frobenius angles: | $\pm0.0885855327829$, $\pm0.287893547303$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $15$ |
Isomorphism classes: | 100 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1800$ | $7711200$ | $22191688800$ | $62279506800000$ | $174887653896945000$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $31$ | $2745$ | $149062$ | $7892993$ | $418195931$ | $22164195510$ | $1174709773247$ | $62259690882913$ | $3299763716578606$ | $174887471925324225$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 15 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=14x^6+38x^5+25x^4+37x^3+27x^2+50x+26$
- $y^2=14x^6+6x^5+35x^4+16x^3+33x^2+6x+4$
- $y^2=37x^5+6x^4+44x^3+25x^2+33x+16$
- $y^2=x^6+42x^5+10x^4+48x^3+12x^2+5x+14$
- $y^2=3x^6+38x^5+39x^4+40x^3+52x^2+50x+21$
- $y^2=x^6+13x^5+45x^4+48x^3+10x^2+22x+51$
- $y^2=30x^6+22x^5+13x^4+22x^3+36x^2+4$
- $y^2=21x^6+34x^5+11x^4+21x^3+32x^2+3x+35$
- $y^2=3x^6+14x^5+4x^4+26x^3+36x^2+19x+52$
- $y^2=47x^6+15x^5+33x^4+41x^3+8x^2+34x+29$
- $y^2=44x^6+3x^5+42x^4+15x^3+8x^2+17x+51$
- $y^2=32x^6+15x^5+8x^4+3x^3+20x^2+3x+51$
- $y^2=20x^6+13x^5+38x^4+48x^3+18x^2+11x+19$
- $y^2=23x^6+17x^5+26x^4+29x^3+44x^2+25x+49$
- $y^2=49x^6+43x^5+43x^4+41x^3+3x^2+24x+44$
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.aj 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.