Invariants
Base field: | $\F_{17}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 8 x + 33 x^{2} - 136 x^{3} + 289 x^{4}$ |
Frobenius angles: | $\pm0.0550704494671$, $\pm0.504752145218$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.39593.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $179$ | $83593$ | $23519168$ | $6890821769$ | $2014054099699$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $292$ | $4786$ | $82500$ | $1418490$ | $24142366$ | $410320186$ | $6975560580$ | $118587959314$ | $2015997159012$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=7x^6+10x^5+5x^4+8x^3+3x^2+5x+2$
- $y^2=3x^6+10x^5+5x^4+6x^3+4x^2+9x+2$
- $y^2=14x^6+7x^5+8x^4+x^3+2x^2+2x+13$
- $y^2=5x^6+14x^5+8x^4+5x^3+x^2+x+11$
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 4.0.39593.1. |
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.i_bh | $2$ | (not in LMFDB) |