Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 14 x^{2} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.182456280080$, $\pm0.817543719920$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{3}, \sqrt{-5})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $24$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $276$ | $76176$ | $24146964$ | $7039881216$ | $2015991481236$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $262$ | $4914$ | $84286$ | $1419858$ | $24156358$ | $410338674$ | $6975799678$ | $118587876498$ | $2015989062022$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 24 curves (of which all are hyperelliptic):
- $y^2=16 x^6+16 x^5+15 x^4+4 x^3+6 x^2+8 x+10$
- $y^2=x^6+x^3+6$
- $y^2=7 x^6+5 x^5+8 x^4+12 x^3+10 x^2+4$
- $y^2=12 x^6+15 x^5+10 x^4+x^3+7 x^2+12 x+9$
- $y^2=2 x^6+11 x^5+13 x^4+3 x^3+4 x^2+2 x+10$
- $y^2=14 x^6+15 x^5+6 x^4+14 x^3+16 x^2+16 x+13$
- $y^2=10 x^6+x^4+8 x^3+11 x^2+8 x+3$
- $y^2=14 x^6+10 x^5+12 x^4+x^3+4 x^2+3 x+15$
- $y^2=6 x^6+2 x^5+6 x^4+11 x^3+4 x^2+10 x+12$
- $y^2=2 x^5+16 x^4+3 x^2+x$
- $y^2=5 x^6+6 x^5+14 x^4+6 x^3+8 x^2+13 x+1$
- $y^2=8 x^6+11 x^5+2 x^4+11 x^3+x+14$
- $y^2=13 x^6+x^5+9 x^4+x^3+16 x^2+15 x+8$
- $y^2=5 x^6+3 x^5+10 x^4+3 x^3+14 x^2+11 x+7$
- $y^2=8 x^6+9 x^5+6 x^4+3 x^3+8 x^2+16 x+7$
- $y^2=13 x^6+7 x^5+5 x^4+13 x^3+3 x^2+5 x+3$
- $y^2=3 x^6+9 x^5+7 x^3+2 x^2+8 x+7$
- $y^2=9 x^6+10 x^5+4 x^3+6 x^2+7 x+4$
- $y^2=15 x^6+x^5+11 x^4+14 x^3+4 x^2+8 x+5$
- $y^2=14 x^5+6 x^4+6 x^3+4 x^2+10 x$
- $y^2=6 x^6+13 x^5+2 x^4+7 x^3+12 x^2+12 x+4$
- $y^2=x^6+5 x^5+6 x^4+4 x^3+2 x^2+2 x+12$
- $y^2=10 x^6+7 x^5+15 x^4+8 x^3+15 x^2+10 x+14$
- $y^2=x^6+x^3+5$
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 \(\Q(\sqrt{3}, \sqrt{-5})\). |
| The base change of $A$ to $\F_{17^{2}}$ is 1.289.ao 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$ |
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.a_o | $4$ | (not in LMFDB) |
| 2.17.am_cn | $12$ | (not in LMFDB) |
| 2.17.m_cn | $12$ | (not in LMFDB) |