Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - x + 13 x^{2} )( 1 + x + 13 x^{2} )$ |
| $1 + 25 x^{2} + 169 x^{4}$ | |
| Frobenius angles: | $\pm0.455715642762$, $\pm0.544284357238$ |
| Angle rank: | $1$ (numerical) |
| Jacobians: | $10$ |
| Isomorphism classes: | 16 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $195$ | $38025$ | $4829760$ | $799475625$ | $137858624475$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $14$ | $220$ | $2198$ | $27988$ | $371294$ | $4832710$ | $62748518$ | $815680228$ | $10604499374$ | $137858757100$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=4 x^6+6 x^5+2 x^4+5 x^3+2 x^2+6 x+4$
- $y^2=8 x^6+12 x^5+4 x^4+10 x^3+4 x^2+12 x+8$
- $y^2=5 x^6+3 x^5+4 x^4+x^3+4 x^2+3 x+5$
- $y^2=10 x^6+6 x^5+8 x^4+2 x^3+8 x^2+6 x+10$
- $y^2=4 x^6+4 x^4+11 x^3+4 x^2+4$
- $y^2=8 x^6+8 x^4+9 x^3+8 x^2+8$
- $y^2=4 x^6+2 x^5+10 x^4+6 x^3+10 x^2+2 x+4$
- $y^2=8 x^6+4 x^5+7 x^4+12 x^3+7 x^2+4 x+8$
- $y^2=12 x^6+6 x^5+9 x^4+2 x^3+9 x^2+6 x+12$
- $y^2=11 x^6+12 x^5+5 x^4+4 x^3+5 x^2+12 x+11$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{2}}$.
Endomorphism algebra over $\F_{13}$| The isogeny class factors as 1.13.ab $\times$ 1.13.b and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
| The base change of $A$ to $\F_{13^{2}}$ is 1.169.z 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$ |
Base change
This is a primitive isogeny class.