Invariants
Base field: | $\F_{17}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 9 x + 41 x^{2} - 153 x^{3} + 289 x^{4}$ |
Frobenius angles: | $\pm0.0511302365929$, $\pm0.466745019538$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.328653.3 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $169$ | $83317$ | $23743993$ | $6893398629$ | $2011960631824$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $9$ | $291$ | $4833$ | $82531$ | $1417014$ | $24139923$ | $410348241$ | $6975598339$ | $118587234249$ | $2015995427166$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=6x^6+3x^5+4x^4+6x^3+14x^2+10x+5$
- $y^2=14x^6+4x^5+11x^4+12x^3+7x^2+3x+7$
- $y^2=4x^6+x^5+16x^4+9x^3+13x^2+x+5$
- $y^2=11x^6+6x^5+3x^4+3x^3+5x^2+11$
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 4.0.328653.3. |
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_bp | $2$ | (not in LMFDB) |