Invariants
Base field: | $\F_{53}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 14 x + 53 x^{2} )( 1 - 12 x + 53 x^{2} )$ |
$1 - 26 x + 274 x^{2} - 1378 x^{3} + 2809 x^{4}$ | |
Frobenius angles: | $\pm0.0885855327829$, $\pm0.191645762723$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $8$ |
Isomorphism classes: | 28 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1680$ | $7539840$ | $22114244880$ | $62273046528000$ | $174901377579104400$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $28$ | $2682$ | $148540$ | $7892174$ | $418228748$ | $22164655914$ | $1174712973644$ | $62259698390686$ | $3299763609603580$ | $174887470340952282$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=14x^6+7x^5+29x^4+50x^3+29x^2+7x+14$
- $y^2=3x^6+37x^5+25x^4+14x^3+25x^2+37x+3$
- $y^2=48x^6+22x^5+23x^4+50x^3+22x^2+5x+45$
- $y^2=10x^6+31x^5+38x^4+35x^3+37x^2+3x+13$
- $y^2=41x^6+35x^5+36x^4+39x^3+24x^2+45x+20$
- $y^2=41x^6+21x^5+24x^4+48x^3+24x^2+21x+41$
- $y^2=22x^6+20x^5+50x^4+6x^3+22x^2+45x+8$
- $y^2=5x^6+6x^5+25x^4+50x^3+25x^2+6x+5$
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.am 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.