Invariants
Base field: | $\F_{13}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 7 x + 13 x^{2} )( 1 - 2 x + 13 x^{2} )$ |
$1 - 9 x + 40 x^{2} - 117 x^{3} + 169 x^{4}$ | |
Frobenius angles: | $\pm0.0772104791556$, $\pm0.410543812489$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $4$ |
Isomorphism classes: | 20 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $84$ | $28224$ | $4826304$ | $806190336$ | $137254909764$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $169$ | $2198$ | $28225$ | $369665$ | $4825798$ | $62765141$ | $815787169$ | $10604499374$ | $137858349889$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=4x^6+6x^5+2x^4+6x^3+3x^2+x+2$
- $y^2=x^6+x^3+11$
- $y^2=11x^6+4x^5+5x^4+2x^3+7x^2+x+2$
- $y^2=7x^6+10x^5+9x^4+6x^3+11x^2+3x+11$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{6}}$.
Endomorphism algebra over $\F_{13}$The isogeny class factors as 1.13.ah $\times$ 1.13.ac 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 $\F_{13^{6}}$ is 1.4826809.atm 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
- Endomorphism algebra over $\F_{13^{2}}$
The base change of $A$ to $\F_{13^{2}}$ is 1.169.ax $\times$ 1.169.w. The endomorphism algebra for each factor is: - Endomorphism algebra over $\F_{13^{3}}$
The base change of $A$ to $\F_{13^{3}}$ is 1.2197.acs $\times$ 1.2197.cs. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.