Invariants
Base field: | $\F_{17}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 5 x + 17 x^{2} )^{2}$ |
$1 - 10 x + 59 x^{2} - 170 x^{3} + 289 x^{4}$ | |
Frobenius angles: | $\pm0.292637436158$, $\pm0.292637436158$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $2$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $169$ | $89401$ | $25441936$ | $7059192361$ | $2016777737689$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $8$ | $308$ | $5174$ | $84516$ | $1420408$ | $24123422$ | $410258584$ | $6975597508$ | $118588438358$ | $2015999428628$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=14x^6+5x^5+8x^4+5x^3+7x^2+14x+6$
- $y^2=8x^6+4x^5+2x^4+5x^3+9x^2+13x+15$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$The isogeny class factors as 1.17.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-43}) \)$)$ |
Base change
This is a primitive isogeny class.