Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 25 x^{2} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.118522015261$, $\pm0.881477984739$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{59})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $9$ |
| Isomorphism classes: | 9 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $265$ | $70225$ | $24143620$ | $6968075625$ | $2015996272825$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $240$ | $4914$ | $83428$ | $1419858$ | $24149670$ | $410338674$ | $6976087108$ | $118587876498$ | $2015998645200$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=10 x^6+7 x^5+15 x^3+14 x+1$
- $y^2=12 x^6+7 x^5+3 x^4+14 x^3+16 x^2+8 x+1$
- $y^2=2 x^6+4 x^5+9 x^4+8 x^3+14 x^2+7 x+3$
- $y^2=7 x^6+13 x^5+14 x^4+11 x^3+13 x^2+5 x+14$
- $y^2=4 x^6+5 x^5+8 x^4+16 x^3+5 x^2+15 x+8$
- $y^2=16 x^6+15 x^5+15 x^4+4 x^3+10 x^2+11 x+3$
- $y^2=14 x^6+11 x^5+11 x^4+12 x^3+13 x^2+16 x+9$
- $y^2=16 x^6+8 x^5+x^4+10 x^2+x+3$
- $y^2=8 x^6+9 x^5+16 x^4+8 x^3+5 x^2+4 x+3$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17^{2}}$.
Endomorphism algebra over $\F_{17}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{59})\). |
| The base change of $A$ to $\F_{17^{2}}$ is 1.289.az 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-59}) \)$)$ |
Base change
This is a primitive isogeny class.