Invariants
| Base field: | $\F_{17}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 11 x + 60 x^{2} - 187 x^{3} + 289 x^{4}$ |
| Frobenius angles: | $\pm0.130625071936$, $\pm0.363090970453$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.43928.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $152$ | $83296$ | $24573536$ | $6986535296$ | $2014729448152$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $7$ | $289$ | $5002$ | $83649$ | $1418967$ | $24136894$ | $410383015$ | $6976082433$ | $118588922026$ | $2015994388929$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=4x^6+11x^5+4x^4+5x^3+3x^2+15x+14$
- $y^2=6x^6+8x^5+5x^4+6x^3+14x^2+11x+11$
- $y^2=4x^6+8x^5+16x^4+3x^3+2x^2+11x+14$
- $y^2=6x^6+11x^5+14x^4+16x^3+5x^2+4x+6$
- $y^2=7x^6+4x^5+4x^4+7x^3+15x^2+7x+12$
- $y^2=5x^6+5x^5+3x^4+x^3+10x^2+7x+12$
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 4.0.43928.1. |
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.l_ci | $2$ | (not in LMFDB) |