Invariants
Base field: | $\F_{17}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 8 x + 17 x^{2} )( 1 - 5 x + 17 x^{2} )$ |
$1 - 13 x + 74 x^{2} - 221 x^{3} + 289 x^{4}$ | |
Frobenius angles: | $\pm0.0779791303774$, $\pm0.292637436158$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $1$ |
Isomorphism classes: | 3 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $130$ | $77740$ | $24261640$ | $6990380800$ | $2015239733650$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $269$ | $4940$ | $83697$ | $1419325$ | $24129506$ | $410304445$ | $6975740833$ | $118588565420$ | $2015998851389$ |
Jacobians and polarizations
This isogeny class contains the Jacobian of 1 curve (which is hyperelliptic), and hence is principally polarizable:
- $y^2=5x^6+5x^5+3x^2+9x+6$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$The isogeny class factors as 1.17.ai $\times$ 1.17.af and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
Base change
This is a primitive isogeny class.