Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 7 x + 13 x^{2} )( 1 - 4 x + 13 x^{2} )$ |
$1 - 11 x + 54 x^{2} - 143 x^{3} + 169 x^{4}$ | |
Frobenius angles: | $\pm0.0772104791556$, $\pm0.312832958189$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $1$ |
Isomorphism classes: | 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $70$ | $26460$ | $4873120$ | $817084800$ | $137610200350$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $157$ | $2220$ | $28609$ | $370623$ | $4822234$ | $62737419$ | $815751841$ | $10604790060$ | $137859735157$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=5x^6+2x^5+3x^4+11x^3+5x^2+3x+5$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$The isogeny class factors as 1.13.ah $\times$ 1.13.ae 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.