Invariants
Base field: | $\F_{17}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 8 x + 35 x^{2} - 136 x^{3} + 289 x^{4}$ |
Frobenius angles: | $\pm0.0961339412965$, $\pm0.495096868820$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.1472400.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $181$ | $84889$ | $23753716$ | $6910728601$ | $2015531053261$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $296$ | $4834$ | $82740$ | $1419530$ | $24149702$ | $410363978$ | $6975716644$ | $118588409218$ | $2015999525336$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=3x^6+x^5+7x^4+x^3+8x^2+14x+11$
- $y^2=8x^6+8x^5+10x^4+6x^3+8x^2+12x+6$
- $y^2=10x^6+14x^5+5x^4+10x^3+11x^2+6$
- $y^2=3x^6+11x^5+6x^4+15x^3+7x^2+4x+10$
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 4.0.1472400.1. |
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_bj | $2$ | (not in LMFDB) |