Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 3 x - 8 x^{2} - 51 x^{3} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.0481446514053$, $\pm0.714811318072$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-3}, \sqrt{-59})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $9$ |
| Isomorphism classes: | 15 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $228$ | $76608$ | $22924944$ | $6979601664$ | $2012756156868$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $15$ | $265$ | $4662$ | $83569$ | $1417575$ | $24125470$ | $410359335$ | $6975592609$ | $118587589974$ | $2015996272825$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 9 curves (of which all are hyperelliptic):
- $y^2=2 x^6+9 x^5+13 x^4+x^3+15 x^2+3 x+2$
- $y^2=7 x^6+x^5+9 x^4+x^3+3 x^2+1$
- $y^2=8 x^6+8 x^5+14 x^4+9 x^2+6 x+8$
- $y^2=4 x^6+2 x^5+3 x^4+15 x^3+3 x^2+9 x+15$
- $y^2=7 x^6+4 x^5+7 x^4+5 x^3+8 x+4$
- $y^2=9 x^5+3 x^4+16 x^3+15 x^2+15 x+4$
- $y^2=12 x^6+7 x^5+x^4+2 x^3+10 x^2+14 x+12$
- $y^2=10 x^6+9 x^5+x^4+11 x^3+11 x^2+6 x$
- $y^2=x^6+16 x^5+15 x^4+x^3+4 x^2+8 x+4$
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.aew 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-59}) \)$)$ |
Base change
This is a primitive isogeny class.