Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 22 x^{2} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.137999402583$, $\pm0.862000597417$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{14})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $10$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $268$ | $71824$ | $24145996$ | $6991635456$ | $2015994945868$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $246$ | $4914$ | $83710$ | $1419858$ | $24154422$ | $410338674$ | $6976073854$ | $118587876498$ | $2015995991286$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=5 x^6+14 x^5+5 x^4+3 x^3+4 x^2+11 x+8$
- $y^2=14 x^6+10 x^5+5 x^4+7 x^3+13 x^2+3 x+15$
- $y^2=7 x^6+16 x^5+8 x^4+8 x^3+11 x^2+9 x+9$
- $y^2=7 x^6+9 x^5+5 x^4+4 x^3+8 x^2+4 x+8$
- $y^2=5 x^6+7 x^5+10 x^4+13 x^3+7 x^2+11 x+8$
- $y^2=2 x^6+7 x^5+3 x^3+13 x^2+10 x+5$
- $y^2=6 x^6+4 x^5+9 x^3+5 x^2+13 x+15$
- $y^2=11 x^6+9 x^5+9 x^4+12 x^3+6 x^2+4 x+2$
- $y^2=16 x^6+6 x^5+4 x^4+5 x^3+6 x^2+5 x+3$
- $y^2=16 x^6+7 x^5+5 x^4+10 x^3+11 x^2+8 x+11$
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 \(\Q(\sqrt{-3}, \sqrt{14})\). |
| The base change of $A$ to $\F_{17^{2}}$ is 1.289.aw 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-42}) \)$)$ |
Base change
This is a primitive isogeny class.