Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 - 7 x + 17 x^{2} )( 1 - 4 x + 17 x^{2} )$ |
| $1 - 11 x + 62 x^{2} - 187 x^{3} + 289 x^{4}$ | |
| Frobenius angles: | $\pm0.177280642489$, $\pm0.338793663197$ |
| Angle rank: | $2$ (numerical) |
| Jacobians: | $4$ |
| Isomorphism classes: | 10 |
| Cyclic group of points: | yes |
This isogeny class is not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $154$ | $84700$ | $24906112$ | $7026712000$ | $2017227550954$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $293$ | $5068$ | $84129$ | $1420727$ | $24137426$ | $410352215$ | $6975902401$ | $118588353436$ | $2015992733093$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=10 x^6+16 x^5+12 x^4+7 x^3+3 x^2+14 x+5$
- $y^2=11 x^6+2 x^5+6 x^4+4 x^3+11 x^2+8 x+16$
- $y^2=15 x^6+7 x^5+5 x^4+12 x^3+13 x^2+14 x+14$
- $y^2=7 x^6+13 x^5+2 x^3+6 x^2+5 x+14$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$| The isogeny class factors as 1.17.ah $\times$ 1.17.ae and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.ad_g | $2$ | (not in LMFDB) |
| 2.17.d_g | $2$ | (not in LMFDB) |
| 2.17.l_ck | $2$ | (not in LMFDB) |