Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 10 x + 51 x^{2} - 190 x^{3} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.0769799514810$, $\pm0.443626041964$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-33 +16 \sqrt{3}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 6 |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $213$ | $130569$ | $46774800$ | $16859199849$ | $6122283767853$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $364$ | $6820$ | $129364$ | $2472550$ | $47050918$ | $893935570$ | $16983615844$ | $322687208860$ | $6131068985404$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=16 x^6+7 x^4+12 x^3+x^2+8 x+4$
- $y^2=8 x^6+6 x^5+13 x^4+11 x^3+14 x^2+3 x+12$
- $y^2=6 x^6+8 x^5+13 x^4+4 x^3+6 x^2+14 x+16$
- $y^2=14 x^6+x^5+16 x^4+8 x^3+11 x^2+13 x+3$
- $y^2=13 x^6+3 x^5+15 x^4+18 x^3+11 x^2+16 x+13$
- $y^2=6 x^6+x^5+16 x^4+17 x^3+x^2+15 x+18$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19}$.
Endomorphism algebra over $\F_{19}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-33 +16 \sqrt{3}})\). |
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 |
|---|---|---|
| 2.19.k_bz | $2$ | (not in LMFDB) |