Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - x + 13 x^{2} )^{3}$ |
| $1 - 3 x + 42 x^{2} - 79 x^{3} + 546 x^{4} - 507 x^{5} + 2197 x^{6}$ | |
| Frobenius angles: | $\pm0.455715642762$, $\pm0.455715642762$, $\pm0.455715642762$ |
| Angle rank: | $1$ (numerical) |
| Cyclic group of points: | no |
| Non-cyclic primes: | $13$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $2197$ | $7414875$ | $11179320256$ | $22605173296875$ | $50863981352595697$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $11$ | $245$ | $2312$ | $27701$ | $368951$ | $4835660$ | $62787827$ | $815654981$ | $10603912616$ | $137858889725$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains no Jacobian of a hyperelliptic curve, but it is unknown whether it contains a Jacobian of a non-hyperelliptic curve.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13}$.
Endomorphism algebra over $\F_{13}$| The isogeny class factors as 1.13.ab 3 and its endomorphism algebra is $\mathrm{M}_{3}($\(\Q(\sqrt{-51}) \)$)$ |
Base change
This is a primitive isogeny class.