Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 5 x + 9 x^{2} - 85 x^{3} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.0620218782481$, $\pm0.622267387770$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-13 +2 \sqrt{5}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
| Cyclic group of points: | yes |
This isogeny class is simple and 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $209$ | $81301$ | $22958441$ | $6939934661$ | $2017271614864$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $13$ | $283$ | $4669$ | $83091$ | $1420758$ | $24127147$ | $410312629$ | $6975922083$ | $118587788053$ | $2015992774878$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=6 x^6+4 x^5+9 x^4+16 x^3+5 x^2+7 x+7$
- $y^2=11 x^6+4 x^4+9 x^3+13 x^2+9 x+14$
- $y^2=7 x^6+12 x^5+5 x^4+3 x^3+6 x^2+16 x+5$
- $y^2=4 x^6+13 x^5+x^4+6 x^3+13 x^2+11 x$
- $y^2=7 x^6+13 x^5+12 x^4+8 x^3+6 x^2+3 x+7$
- $y^2=x^6+16 x^5+3 x^4+3 x^3+2 x^2+12 x+9$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-13 +2 \sqrt{5}})\). |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
| Twist | Extension degree | Common base change |
|---|---|---|
| 2.17.f_j | $2$ | (not in LMFDB) |