Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 7 x + 13 x^{2} )( 1 + 13 x^{2} )$ |
$1 - 7 x + 26 x^{2} - 91 x^{3} + 169 x^{4}$ | |
Frobenius angles: | $\pm0.0772104791556$, $\pm0.5$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $2$ |
Isomorphism classes: | 18 |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $98$ | $28812$ | $4677344$ | $800743104$ | $137700691898$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $7$ | $173$ | $2128$ | $28033$ | $370867$ | $4830698$ | $62750527$ | $815694241$ | $10604617744$ | $137859794693$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+3x^5+5x^3+12x^2+2x+11$
- $y^2=8x^6+3x^5+3x^4+3x^3+11x^2+7x+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{2}}$.
Endomorphism algebra over $\F_{13}$The isogeny class factors as 1.13.ah $\times$ 1.13.a and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
The base change of $A$ to $\F_{13^{2}}$ is 1.169.ax $\times$ 1.169.ba. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.