Invariants
Base field: | $\F_{17}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 12 x + 67 x^{2} - 204 x^{3} + 289 x^{4}$ |
Frobenius angles: | $\pm0.112997812760$, $\pm0.326838618982$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.58896.2 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $141$ | $80793$ | $24491700$ | $6996108249$ | $2015297698821$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $6$ | $280$ | $4986$ | $83764$ | $1419366$ | $24133030$ | $410345718$ | $6975975844$ | $118589241402$ | $2015998395400$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=9x^6+15x^5+2x^4+9x^2+16x+14$
- $y^2=5x^6+15x^5+3x^4+2x^3+x^2+8x+10$
- $y^2=10x^6+5x^5+14x^4+15x^3+14x^2+14x+6$
- $y^2=6x^6+x^5+10x^4+2x^3+9x^2+4x+5$
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 4.0.58896.2. |
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.m_cp | $2$ | (not in LMFDB) |