Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 8 x + 52 x^{2} + 152 x^{3} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.595856155308$, $\pm0.713294793489$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.207104.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $10$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $574$ | $145796$ | $45158302$ | $17034221456$ | $6137229697774$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $28$ | $402$ | $6580$ | $130710$ | $2478588$ | $47033922$ | $893880148$ | $16983599454$ | $322687889020$ | $6131065874002$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 10 curves (of which all are hyperelliptic):
- $y^2=6 x^6+10 x^4+9 x^3+12 x^2+7 x+11$
- $y^2=17 x^6+x^5+2 x^4+13 x^3+6 x^2+4 x+18$
- $y^2=x^5+5 x^4+x^3+9 x^2+4 x+8$
- $y^2=4 x^6+8 x^5+4 x^4+7 x^3+10 x^2+x+15$
- $y^2=9 x^6+4 x^5+13 x^4+9 x^3+13 x^2+10 x+5$
- $y^2=9 x^6+6 x^5+14 x^4+4 x^3+6 x^2+9 x+16$
- $y^2=9 x^6+x^5+18 x^4+10 x^3+14 x^2+12 x+12$
- $y^2=13 x^6+15 x^5+14 x^4+13 x^3+15 x+6$
- $y^2=9 x^6+3 x^5+8 x^4+3 x^3+7 x^2+14 x+1$
- $y^2=6 x^6+4 x^5+11 x^4+13 x^3+17 x^2+1$
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 4.0.207104.1. |
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 |
|---|---|---|
| 2.19.ai_ca | $2$ | (not in LMFDB) |