Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 + 47 x^{2} + 4 x^{3} + 893 x^{4} + 6859 x^{6}$ |
| Frobenius angles: | $\pm0.390535910551$, $\pm0.485143875220$, $\pm0.625387659730$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.263396610560.1 |
| Galois group: | $S_4\times C_2$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $7804$ | $60839984$ | $323241859492$ | $2199010116093440$ | $15175360254530746924$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $456$ | $6872$ | $129476$ | $2475160$ | $47042808$ | $893908588$ | $16983822012$ | $322686735332$ | $6131062486616$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 17 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=x^8+13 x^7+5 x^6+8 x^5+3 x^4+13 x^3+6 x^2+3 x+1$
- $y^2=x^8+x^7+18 x^5+3 x^4+14 x^2+9 x$
- $y^2=x^8+8 x^7+9 x^6+15 x^5+17 x^4+10 x^3+12 x+5$
- $y^2=x^8+14 x^7+14 x^6+2 x^4+9 x^3+2 x^2+5 x+17$
- $y^2=18 x^8+15 x^7+10 x^6+13 x^5+7 x^4+7 x^3+15 x^2+9 x+1$
- $y^2=18 x^8+2 x^7+x^6+13 x^5+10 x^4+x^3+10 x^2+9 x+5$
- $y^2=18 x^7+8 x^6+3 x^5+3 x^4+10 x^3+3 x^2+12 x+2$
- $y^2=18 x^8+13 x^7+x^6+17 x^5+3 x^4+18 x^2+7 x+6$
- $y^2=18 x^8+5 x^7+14 x^6+17 x^5+4 x^4+7 x^3+12 x^2+3 x+2$
- $y^2=x^8+5 x^7+2 x^6+5 x^5+18 x^4+2 x^3+3 x^2+x+4$
- $y^2=x^8+16 x^6+17 x^5+2 x^4+6 x^3+14 x^2+17 x+9$
- $y^2=18 x^8+16 x^7+13 x^6+3 x^5+14 x^4+x^3+12 x^2+18 x+15$
- $y^2=18 x^8+5 x^7+9 x^5+6 x^3+14 x^2+18 x+7$
- $y^2=x^8+15 x^7+7 x^6+7 x^5+5 x^4+10 x^3+13 x^2+11 x+1$
- $y^2=18 x^8+10 x^7+14 x^6+14 x^5+17 x^4+7 x^3+3 x^2+11 x+16$
- $y^2=18 x^8+4 x^7+4 x^6+8 x^5+14 x^4+15 x^3+3 x^2+8 x+17$
- $y^2=x^8+14 x^7+3 x^6+12 x^5+17 x^4+11 x^3+9 x^2+8 x+6$
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 6.0.263396610560.1. |
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.a_bv_ae | $2$ | (not in LMFDB) |