Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 12 x + 68 x^{2} + 228 x^{3} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.633519705146$, $\pm0.920823034645$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-34 +12 \sqrt{6}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $4$ |
| Isomorphism classes: | 4 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $670$ | $127300$ | $46793470$ | $16942611600$ | $6140314766350$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $32$ | $354$ | $6824$ | $130006$ | $2479832$ | $47033682$ | $893849408$ | $16983923806$ | $322686321536$ | $6131067736674$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 4 curves (of which all are hyperelliptic):
- $y^2=6 x^6+12 x^5+11 x^4+4 x^3+3 x^2+16 x+13$
- $y^2=9 x^6+2 x^5+10 x^4+4 x^3+14 x^2+5 x+15$
- $y^2=8 x^6+17 x^5+8 x^4+7 x^3+8 x^2+4 x+10$
- $y^2=x^6+7 x^5+16 x^4+15 x^3+11 x+4$
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 \(\Q(\sqrt{-34 +12 \sqrt{6}})\). |
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.am_cq | $2$ | (not in LMFDB) |