Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 14 x^{2} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.159502860020$, $\pm0.840497139980$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{10})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $10$ |
| Isomorphism classes: | 40 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $156$ | $24336$ | $4831164$ | $823919616$ | $137858273436$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $14$ | $142$ | $2198$ | $28846$ | $371294$ | $4835518$ | $62748518$ | $815804638$ | $10604499374$ | $137858055022$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=x^5+6 x^4+4 x^3+3 x^2+10 x$
- $y^2=10 x^6+8 x^5+2 x^4+7 x^3+3 x+4$
- $y^2=3 x^6+x^5+10 x^4+x^3+8 x^2+10 x+11$
- $y^2=3 x^6+11 x^5+11 x^4+2 x^3+10 x^2+12 x+6$
- $y^2=6 x^6+9 x^5+9 x^4+4 x^3+7 x^2+11 x+12$
- $y^2=x^6+3 x^5+3 x^4+7 x^3+3 x^2+5 x$
- $y^2=x^6+x^3+5$
- $y^2=x^6+12 x^4+8 x^3+6 x^2+5$
- $y^2=x^6+x^3+8$
- $y^2=10 x^6+12 x^5+10 x^4+x^3+5 x^2+3 x+11$
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 \(\Q(\sqrt{-3}, \sqrt{10})\). |
| The base change of $A$ to $\F_{13^{2}}$ is 1.169.ao 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-30}) \)$)$ |
Base change
This is a primitive isogeny class.