Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 3 x - 8 x^{2} + 51 x^{3} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.285188681928$, $\pm0.951855348595$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-59})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $9$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $336$ | $76608$ | $25401600$ | $6979601664$ | $2019239228496$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $21$ | $265$ | $5166$ | $83569$ | $1422141$ | $24125470$ | $410318013$ | $6975592609$ | $118588163022$ | $2015996272825$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=6 x^6+10 x^5+5 x^4+3 x^3+11 x^2+9 x+6$
- $y^2=4 x^6+3 x^5+10 x^4+3 x^3+9 x^2+3$
- $y^2=7 x^6+7 x^5+8 x^4+10 x^2+x+7$
- $y^2=12 x^6+6 x^5+9 x^4+11 x^3+9 x^2+10 x+11$
- $y^2=8 x^6+7 x^5+8 x^4+13 x^3+14 x+7$
- $y^2=10 x^5+9 x^4+14 x^3+11 x^2+11 x+12$
- $y^2=4 x^6+8 x^5+6 x^4+12 x^3+9 x^2+16 x+4$
- $y^2=13 x^6+10 x^5+3 x^4+16 x^3+16 x^2+x$
- $y^2=3 x^6+14 x^5+11 x^4+3 x^3+12 x^2+7 x+12$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17^{3}}$.
Endomorphism algebra over $\F_{17}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{-59})\). |
| The base change of $A$ to $\F_{17^{3}}$ is 1.4913.ew 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-59}) \)$)$ |
Base change
This is a primitive isogeny class.