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: | 4.0.969024.4 |
| Galois group: | $D_{4}$ |
| Jacobians: | $8$ |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $218$ | $134724$ | $47812850$ | $16980074064$ | $6129701550578$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $10$ | $374$ | $6970$ | $130294$ | $2475550$ | $47058518$ | $893983870$ | $16983882334$ | $322687210810$ | $6131060250854$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=13x^6+7x^5+4x^4+5x^3+17x^2+13x+1$
- $y^2=6x^6+14x^5+x^4+13x^3+8x^2+5x+16$
- $y^2=12x^6+2x^5+12x^4+9x^3+16x^2+6x+12$
- $y^2=x^6+9x^5+16x^4+7x^3+4x^2+16x+4$
- $y^2=18x^6+17x^5+x^4+5x^3+5x+8$
- $y^2=8x^6+x^5+6x^4+3x^3+15x^2+3x+12$
- $y^2=3x^6+3x^5+12x^4+17x^3+15x^2+15$
- $y^2=7x^6+10x^5+16x^4+17x^3+x^2+18x+12$
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 4.0.969024.4. |
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) |