Invariants
Base field: | $\F_{17}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 9 x + 43 x^{2} - 153 x^{3} + 289 x^{4}$ |
Frobenius angles: | $\pm0.0985547503741$, $\pm0.455623877170$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.10525.1 |
Galois group: | $D_{4}$ |
Jacobians: | $15$ |
Isomorphism classes: | 21 |
This isogeny class is simple and geometrically 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $171$ | $84645$ | $24009939$ | $6919305525$ | $2013747767856$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $9$ | $295$ | $4887$ | $82843$ | $1418274$ | $24146935$ | $410394987$ | $6975815443$ | $118587867669$ | $2015997236350$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 15 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=14x^6+5x^5+15x^4+x^3+5x^2+7x+11$
- $y^2=x^6+9x^5+x^4+14x^3+15x^2+2x+14$
- $y^2=3x^6+10x^5+16x^4+14x^3+3x^2+10x+5$
- $y^2=10x^6+3x^3+13x^2+9x+11$
- $y^2=10x^6+16x^5+11x^4+x^3+8x^2+6x+13$
- $y^2=5x^6+12x^5+4x^4+x^3+7x^2+13x+14$
- $y^2=7x^6+4x^5+13x^4+6x^3+16x^2+4x+7$
- $y^2=10x^6+14x^5+7x^4+5x^3+11x^2+16x+10$
- $y^2=12x^6+9x^5+8x^4+7x^3+8x^2+10x+4$
- $y^2=9x^6+2x^5+4x^4+11x^3+7x^2+9x+6$
- $y^2=14x^6+15x^5+2x^4+14x^2+8x+2$
- $y^2=6x^5+11x^4+x^3+6x^2+10x+12$
- $y^2=15x^6+x^5+3x^4+14x^3+11x^2+11x+3$
- $y^2=15x^5+3x^4+x^3+3x^2+2x+3$
- $y^2=7x^6+x^5+x^4+12x^3+5x^2+13x+6$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{17}$.
Endomorphism algebra over $\F_{17}$The endomorphism algebra of this simple isogeny class is 4.0.10525.1. |
Base change
This is a primitive isogeny class.
Twists
Below is a list of all twists of this isogeny class.
Twist | Extension degree | Common base change |
---|---|---|
2.17.j_br | $2$ | (not in LMFDB) |