Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 3 x + 25 x^{2} - 51 x^{3} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.299661076636$, $\pm0.572186887130$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-218 +18 \sqrt{5}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $20$ |
| Isomorphism classes: | 28 |
| Cyclic group of points: | no |
| Non-cyclic primes: | $3$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $261$ | $96309$ | $24356781$ | $6985195461$ | $2019080699136$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $15$ | $331$ | $4959$ | $83635$ | $1422030$ | $24131707$ | $410259543$ | $6975744739$ | $118588889943$ | $2015994917086$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which all are hyperelliptic):
- $y^2=9 x^6+13 x^5+6 x^4+11 x^3+8 x^2+10 x+7$
- $y^2=6 x^6+9 x^5+15 x^4+8 x^3+13 x^2+10$
- $y^2=11 x^5+x^4+4 x^3+8 x^2+3 x+8$
- $y^2=6 x^6+14 x^5+5 x^4+3 x^3+5 x^2+10 x+16$
- $y^2=3 x^6+9 x^5+11 x^4+8 x^3+9 x^2+11 x+11$
- $y^2=5 x^6+6 x^5+x^4+x^3+7 x^2+4$
- $y^2=3 x^6+7 x^5+11 x^4+4 x^3+11 x^2+8 x+4$
- $y^2=9 x^6+16 x^5+6 x^4+14 x^3+16 x^2+11 x+8$
- $y^2=3 x^6+2 x^5+11 x^4+11 x^3+8 x$
- $y^2=x^6+16 x^5+4 x^4+13 x^3+9 x^2+14 x+11$
- $y^2=16 x^6+10 x^5+15 x^4+13 x^3+5 x^2+8 x+15$
- $y^2=4 x^6+5 x^5+6 x^4+9 x^3+11 x^2+15 x+5$
- $y^2=16 x^5+14 x^4+8 x^3+6 x^2+2 x+8$
- $y^2=6 x^6+14 x^5+15 x^4+2 x^3+4 x^2+3 x+2$
- $y^2=x^6+11 x^5+4 x^4+6 x^3+x+11$
- $y^2=11 x^6+13 x^5+12 x^4+15 x^3+15 x+13$
- $y^2=3 x^6+15 x^5+7 x^4+15 x^3+9 x^2+8 x+10$
- $y^2=11 x^6+16 x^5+14 x^4+5 x^3+8 x^2+8$
- $y^2=12 x^6+6 x^5+6 x^4+6 x^3+13 x^2+14 x+15$
- $y^2=11 x^6+11 x^5+6 x^4+7 x^3+3 x^2+12 x+4$
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 \(\Q(\sqrt{-218 +18 \sqrt{5}})\). |
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.d_z | $2$ | (not in LMFDB) |