Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 22 x^{2} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.0894561875110$, $\pm0.910543812489$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\zeta_{12})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $5$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $148$ | $21904$ | $4827316$ | $807469056$ | $137859194068$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $14$ | $126$ | $2198$ | $28270$ | $371294$ | $4827822$ | $62748518$ | $815802334$ | $10604499374$ | $137859896286$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 5 curves (of which all are hyperelliptic):
- $y^2=6 x^6+4 x^5+11 x^4+3 x^3+12 x^2+x+4$
- $y^2=12 x^5+7 x^3+3 x$
- $y^2=x^6+6 x^5+4 x^4+x^2+11 x+12$
- $y^2=x^6+9 x^5+6 x^4+11 x^3+9 x^2+4 x+5$
- $y^2=9 x^6+9 x^5+7 x^4+5 x^2+10 x+4$
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.aw 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-3}) \)$)$ |
Base change
This is a primitive isogeny class.