Invariants
Base field: | $\F_{17}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 8 x + 17 x^{2} )( 1 - 2 x + 17 x^{2} )$ |
$1 - 10 x + 50 x^{2} - 170 x^{3} + 289 x^{4}$ | |
Frobenius angles: | $\pm0.0779791303774$, $\pm0.422020869623$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $10$ |
Isomorphism classes: | 43 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $160$ | $83200$ | $24088480$ | $6922240000$ | $2011667984800$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $8$ | $290$ | $4904$ | $82878$ | $1416808$ | $24137570$ | $410384584$ | $6975884158$ | $118587729608$ | $2015993900450$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^5+12x^4+8x^3+7x^2+10x+6$
- $y^2=6x^6+7x^5+3x^4+12x^3+12x^2+10x+10$
- $y^2=6x^6+2x^5+11x^4+11x^2+15x+6$
- $y^2=10x^6+11x^5+6x^4+x^3+10x^2+13$
- $y^2=3x^6+13x^5+x^4+6x^3+2x^2+12x+7$
- $y^2=14x^6+2x^5+11x^4+12x^3+10x^2+5x+4$
- $y^2=6x^6+13x^5+x^4+13x^2+2x+9$
- $y^2=14x^6+x^5+11x^4+11x^3+6x^2+x+5$
- $y^2=12x^6+14x^5+16x^4+x^3+5x^2+5x+10$
- $y^2=3x^6+10x^5+10x^4+3x^3+12x^2+11x+6$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17^{4}}$.
Endomorphism algebra over $\F_{17}$The isogeny class factors as 1.17.ai $\times$ 1.17.ac and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
The base change of $A$ to $\F_{17^{4}}$ is 1.83521.amk 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$ |
- Endomorphism algebra over $\F_{17^{2}}$
The base change of $A$ to $\F_{17^{2}}$ is 1.289.abe $\times$ 1.289.be. The endomorphism algebra for each factor is:
Base change
This is a primitive isogeny class.