Invariants
Base field: | $\F_{17}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 6 x + 17 x^{2} )^{2}$ |
$1 - 12 x + 70 x^{2} - 204 x^{3} + 289 x^{4}$ | |
Frobenius angles: | $\pm0.240632536990$, $\pm0.240632536990$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $6$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $144$ | $82944$ | $25040016$ | $7072137216$ | $2021435619984$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $286$ | $5094$ | $84670$ | $1423686$ | $24141022$ | $410294310$ | $6975432574$ | $118586681478$ | $2015992253086$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=3x^5+14x$
- $y^2=14x^6+16x^5+13x^4+14x^3+x^2+x+12$
- $y^2=12x^6+15x^5+15x^4+x^3+x^2+8x+7$
- $y^2=16x^6+12x^4+12x^2+16$
- $y^2=9x^6+13x^4+13x^2+9$
- $y^2=10x^6+5x^5+13x^4+5x^3+9x^2+3x+12$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$The isogeny class factors as 1.17.ag 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-2}) \)$)$ |
Base change
This is a primitive isogeny class.