Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 10 x + 56 x^{2} - 190 x^{3} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.159522866725$, $\pm0.412959486295$ |
| Angle rank: | $2$ (numerical) |
| Number field: | \(\Q(\sqrt{-44 -10 \sqrt{7}})\) |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $218$ | $134724$ | $47812850$ | $16980074064$ | $6129701550578$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $374$ | $6970$ | $130294$ | $2475550$ | $47058518$ | $893983870$ | $16983882334$ | $322687210810$ | $6131060250854$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 8 curves (of which all are hyperelliptic):
- $y^2=13 x^6+7 x^5+4 x^4+5 x^3+17 x^2+13 x+1$
- $y^2=6 x^6+14 x^5+x^4+13 x^3+8 x^2+5 x+16$
- $y^2=12 x^6+2 x^5+12 x^4+9 x^3+16 x^2+6 x+12$
- $y^2=x^6+9 x^5+16 x^4+7 x^3+4 x^2+16 x+4$
- $y^2=18 x^6+17 x^5+x^4+5 x^3+5 x+8$
- $y^2=8 x^6+x^5+6 x^4+3 x^3+15 x^2+3 x+12$
- $y^2=3 x^6+3 x^5+12 x^4+17 x^3+15 x^2+15$
- $y^2=7 x^6+10 x^5+16 x^4+17 x^3+x^2+18 x+12$
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 \(\Q(\sqrt{-44 -10 \sqrt{7}})\). |
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_ce | $2$ | (not in LMFDB) |