Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - x + 13 x^{2} )^{2}$ |
| $1 - 2 x + 27 x^{2} - 26 x^{3} + 169 x^{4}$ | |
| Frobenius angles: | $\pm0.455715642762$, $\pm0.455715642762$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $6$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $13$ |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $169$ | $38025$ | $4999696$ | $799475625$ | $137279883169$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $12$ | $220$ | $2274$ | $27988$ | $369732$ | $4832710$ | $62774724$ | $815680228$ | $10604108202$ | $137858757100$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=2 x^6+2 x^3+2$
- $y^2=6 x^6+8 x^5+x^4+8 x^3+x^2+8 x+6$
- $y^2=11 x^6+6 x^5+3 x^4+9 x^3+x^2+5 x+11$
- $y^2=x^6+9 x^5+10 x^4+9 x^3+4 x^2+3 x+1$
- $y^2=4 x^6+6 x^5+2 x^4+10 x^3+5 x^2+5 x+4$
- $y^2=7 x^6+3 x^5+10 x^4+5 x^3+12 x^2+4 x+5$
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 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$ |
Base change
This is a primitive isogeny class.