Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 3 x + 51 x^{2} - 97 x^{3} + 969 x^{4} - 1083 x^{5} + 6859 x^{6}$ |
| Frobenius angles: | $\pm0.379314590909$, $\pm0.419367872965$, $\pm0.589212688586$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.239023791.1 |
| Galois group: | $A_4\times C_2$ |
| 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: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $6697$ | $60694911$ | $329355908443$ | $2200901928801831$ | $15163445775170779687$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $17$ | $455$ | $7001$ | $129587$ | $2473217$ | $47036183$ | $893903048$ | $16984098995$ | $322687798631$ | $6131053581035$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 13 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^7+14 x^5+6 x^4+2 x^3+14 x^2+17 x+5$
- $y^2=18 x^7+7 x^5+14 x^4+13 x^3+4 x^2+7 x+10$
- $y^2=x^7+2 x^5+8 x^4+8 x^3+11 x^2+12 x+9$
- $y^2=18 x^7+14 x^5+3 x^4+15 x^3+8 x^2+x+6$
- $y^2=x^7+12 x^5+3 x^4+3 x^3+13 x^2+7 x+10$
- $y^2=18 x^7+9 x^4+12 x^3+7 x^2+18 x+7$
- $y^2=18 x^7+10 x^5+4 x^4+8 x^3+17 x^2+11 x+2$
- $y^2=x^7+10 x^4+8 x^3+16 x^2+16 x+9$
- $y^2=18 x^7+17 x^5+18 x^4+11 x^3+4 x^2+6 x+18$
- $y^2=x^7+9 x^4+7 x^3+x^2+8 x+3$
- $y^2=x^7+8 x^5+17 x^4+x^3+2 x^2+16 x+6$
- $y^2=18 x^7+8 x^5+16 x^4+8 x^3+10 x^2+18 x+3$
- $y^2=x^7+8 x^4+17 x^3+4 x^2+4 x+12$
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 6.0.239023791.1. |
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 |
|---|---|---|
| 3.19.d_bz_dt | $2$ | (not in LMFDB) |