Invariants
Base field: | $\F_{17}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 10 x + 52 x^{2} - 170 x^{3} + 289 x^{4}$ |
Frobenius angles: | $\pm0.122222492446$, $\pm0.407842140074$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.467264.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $162$ | $84564$ | $24386994$ | $6954881616$ | $2013794032482$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $8$ | $294$ | $4964$ | $83270$ | $1418308$ | $24142518$ | $410411464$ | $6976037374$ | $118588179848$ | $2015993321814$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=5x^6+13x^5+13x^4+8x^3+16x^2+14x+10$
- $y^2=7x^6+6x^5+13x^4+x^3+15x^2+10x+4$
- $y^2=14x^6+9x^5+2x^4+14x^3+x^2+12x+14$
- $y^2=3x^5+7x^4+7x^2+4x+3$
- $y^2=6x^6+9x^5+5x^4+4x^3+9x^2+6x+6$
- $y^2=5x^6+5x^5+11x^4+11x^3+4x^2+5x+5$
- $y^2=10x^6+6x^5+2x^4+7x^2+3x+14$
- $y^2=14x^6+5x^5+9x^4+3x^3+x^2+4x+10$
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 4.0.467264.1. |
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.k_ca | $2$ | (not in LMFDB) |