Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $3$ |
| L-polynomial: | $1 - 12 x + 93 x^{2} - 460 x^{3} + 1767 x^{4} - 4332 x^{5} + 6859 x^{6}$ |
| Frobenius angles: | $\pm0.217030023492$, $\pm0.300380913555$, $\pm0.495782070627$ |
| Angle rank: | $3$ (numerical) |
| Number field: | 6.0.16329926592.1 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $3916$ | $52959984$ | $334215683956$ | $2222164492171008$ | $15198359773327480156$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $8$ | $404$ | $7100$ | $130844$ | $2478908$ | $47056388$ | $893816960$ | $16983049532$ | $322687116368$ | $6131071858244$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 23 hyperelliptic curves, but it is unknown how many Jacobians of non-hyperelliptic curves it contains:
- $y^2=18 x^8+x^7+4 x^6+17 x^5+11 x^4+10 x^3+18 x^2+4 x+12$
- $y^2=18 x^8+10 x^7+9 x^6+7 x^5+6 x^4+13 x^3+2 x^2+10 x+15$
- $y^2=18 x^8+13 x^7+8 x^6+12 x^5+15 x^4+8 x^3+5 x^2+13 x+18$
- $y^2=18 x^8+17 x^7+16 x^6+12 x^5+4 x^4+6 x^3+3 x^2+18 x+2$
- $y^2=18 x^8+x^7+4 x^6+3 x^5+x^4+13 x^3+3 x^2+2 x+14$
- $y^2=18 x^8+4 x^7+16 x^6+12 x^5+x^4+2 x^3+6 x^2+5 x+15$
- $y^2=18 x^8+2 x^7+3 x^6+17 x^5+18 x^4+3 x^3+x^2+15 x+13$
- $y^2=18 x^8+15 x^7+17 x^6+12 x^4+18 x^3+9 x^2+10 x+6$
- $y^2=18 x^8+x^7+8 x^6+4 x^5+2 x^4+10 x^3+12 x^2+9 x+14$
- $y^2=18 x^8+5 x^7+7 x^6+4 x^4+13 x^3+15 x^2+6 x+14$
- $y^2=18 x^8+2 x^6+7 x^5+8 x^4+15 x^3+3 x^2+18 x+17$
- $y^2=x^8+11 x^7+14 x^6+x^4+12 x^3+14 x^2+14 x+3$
- $y^2=18 x^8+11 x^7+13 x^6+13 x^5+17 x^4+17 x^3+9 x^2+14 x+10$
- $y^2=18 x^8+x^7+8 x^6+3 x^5+7 x^4+13 x^3+17 x^2+16 x+15$
- $y^2=18 x^8+14 x^7+8 x^6+18 x^5+2 x^4+13 x^3+8 x^2+15 x+13$
- $y^2=x^8+16 x^7+18 x^6+12 x^5+6 x^4+x^3+3 x^2+11 x+3$
- $y^2=18 x^8+13 x^7+x^6+18 x^5+x^4+7 x^3+13 x+8$
- $y^2=18 x^8+16 x^7+2 x^6+3 x^5+11 x^3+5 x^2+8 x+15$
- $y^2=18 x^8+6 x^7+5 x^6+5 x^5+16 x^4+14 x^3+18 x^2+2 x+2$
- $y^2=18 x^8+15 x^7+2 x^6+10 x^5+16 x^4+7 x^3+4 x^2+16 x+1$
- $y^2=x^8+14 x^6+11 x^5+17 x^4+15 x^3+3 x^2+18 x+12$
- $y^2=x^8+x^7+9 x^6+2 x^5+7 x^4+16 x^3+11 x^2+17 x+3$
- $y^2=x^8+9 x^7+3 x^6+7 x^5+8 x^4+14 x^3+3 x^2+x+5$
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 6.0.16329926592.1. |
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.m_dp_rs | $2$ | (not in LMFDB) |