Invariants
Base field: | $\F_{17}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 8 x + 17 x^{2} )( 1 - 4 x + 17 x^{2} )$ |
$1 - 12 x + 66 x^{2} - 204 x^{3} + 289 x^{4}$ | |
Frobenius angles: | $\pm0.0779791303774$, $\pm0.338793663197$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $4$ |
Isomorphism classes: | 12 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $140$ | $80080$ | $24309740$ | $6970163200$ | $2012913910700$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $278$ | $4950$ | $83454$ | $1417686$ | $24126806$ | $410328582$ | $6975923326$ | $118588965030$ | $2015997066518$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=7x^6+15x^5+4x^4+4x^3+4x^2+15x+7$
- $y^2=5x^6+x^5+16x^3+x+5$
- $y^2=11x^6+10x^5+14x^4+13x^3+10x^2+2x+1$
- $y^2=3x^6+5x^5+x^3+9x^2+15x+15$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$The isogeny class factors as 1.17.ai $\times$ 1.17.ae 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.