Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - x^{2} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.243877145822$, $\pm0.756122854178$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\zeta_{12})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $12$ |
| Cyclic group of points: | yes |
This isogeny class is simple but not geometrically 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $169$ | $28561$ | $4827316$ | $835152201$ | $137858349889$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $14$ | $168$ | $2198$ | $29236$ | $371294$ | $4827822$ | $62748518$ | $815617828$ | $10604499374$ | $137858207928$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=8 x^6+11 x^5+10 x^4+6 x^3+11$
- $y^2=3 x^6+9 x^5+7 x^4+12 x^3+9$
- $y^2=3 x^6+2 x^5+12 x^3+8 x+2$
- $y^2=12 x^6+10 x^5+5 x^4+x^3+10 x^2+7 x+11$
- $y^2=11 x^6+7 x^5+10 x^4+2 x^3+7 x^2+x+9$
- $y^2=6 x^6+10 x^5+x^2+4$
- $y^2=11 x^6+5 x^5+6 x^4+7 x^3+x^2+7 x+10$
- $y^2=9 x^6+10 x^5+12 x^4+x^3+2 x^2+x+7$
- $y^2=4 x^6+x^5+11 x^4+2 x^3+6 x^2+10$
- $y^2=8 x^6+2 x^5+9 x^4+4 x^3+12 x^2+7$
- $y^2=9 x^6+12 x^5+12 x^4+5 x^3+2 x^2+9 x+6$
- $y^2=7 x^6+10 x^5+8 x^4+5 x^3+10 x^2+x+9$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{2}}$.
Endomorphism algebra over $\F_{13}$| The endomorphism algebra of this simple isogeny class is \(\Q(\zeta_{12})\). |
| The base change of $A$ to $\F_{13^{2}}$ is 1.169.ab 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This is a primitive isogeny class.