Invariants
| Base field: | $\F_{13}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + x - 12 x^{2} + 13 x^{3} + 169 x^{4}$ |
| Frobenius angles: | $\pm0.210951023905$, $\pm0.877617690571$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{17})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $10$ |
| Isomorphism classes: | 18 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $172$ | $24768$ | $4999696$ | $823981824$ | $138148711132$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $15$ | $145$ | $2274$ | $28849$ | $372075$ | $4832710$ | $62735415$ | $815755969$ | $10604108202$ | $137858359225$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=2 x^6+4 x^3+8$
- $y^2=6 x^6+9 x^5+10 x^4+8 x^3+8 x^2+6 x+12$
- $y^2=8 x^6+7 x^4+x^3+9 x^2+x+10$
- $y^2=5 x^6+11 x^5+4 x^3+10 x^2+x+1$
- $y^2=6 x^6+6 x^5+12 x^4+8 x^3+12 x^2+9$
- $y^2=11 x^6+11 x^5+5 x^4+3 x^3+x^2+3 x$
- $y^2=x^6+11 x^5+7 x^4+6 x^2+6 x+1$
- $y^2=12 x^6+6 x^5+6 x^4+9 x^3+2 x^2+4 x+1$
- $y^2=5 x^6+4 x^5+x^4+2 x^3+7 x^2+9 x+7$
- $y^2=7 x^6+10 x^5+4 x^4+x^2+9 x+4$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{13^{3}}$.
Endomorphism algebra over $\F_{13}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{17})\). |
| The base change of $A$ to $\F_{13^{3}}$ is 1.2197.bm 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-51}) \)$)$ |
Base change
This is a primitive isogeny class.