Invariants
Base field: | $\F_{13}$ |
Dimension: | $3$ |
L-polynomial: | $( 1 - 2 x + 13 x^{2} )( 1 - 7 x + 13 x^{2} )^{2}$ |
$1 - 16 x + 116 x^{2} - 514 x^{3} + 1508 x^{4} - 2704 x^{5} + 2197 x^{6}$ | |
Frobenius angles: | $\pm0.0772104791556$, $\pm0.0772104791556$, $\pm0.410543812489$ |
Angle rank: | $1$ (numerical) |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
$p$-rank: | $3$ |
Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $588$ | $4148928$ | $10270374912$ | $22872426022656$ | $50903316619445388$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-2$ | $146$ | $2128$ | $28034$ | $369238$ | $4825292$ | $62767150$ | $815807810$ | $10604617744$ | $137858910146$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
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 2 $\times$ 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 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-3}) \)$)$ |
- Endomorphism algebra over $\F_{13^{2}}$
The base change of $A$ to $\F_{13^{2}}$ is 1.169.ax 2 $\times$ 1.169.w. The endomorphism algebra for each factor is: - 1.169.ax 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 1.169.w : \(\Q(\sqrt{-3}) \).
- Endomorphism algebra over $\F_{13^{3}}$
The base change of $A$ to $\F_{13^{3}}$ is 1.2197.acs 2 $\times$ 1.2197.cs. The endomorphism algebra for each factor is: - 1.2197.acs 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$
- 1.2197.cs : \(\Q(\sqrt{-3}) \).
Base change
This is a primitive isogeny class.