Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $( 1 + 19 x^{2} )^{2}$ |
| $1 + 38 x^{2} + 361 x^{4}$ | |
| Frobenius angles: | $\pm0.5$, $\pm0.5$ |
| Angle rank: | $0$ (numerical) |
| Jacobians: | $12$ |
| Cyclic group of points: | no |
| Non-cyclic primes: | $2, 5$ |
This isogeny class is not simple, primitive, not ordinary, and supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is supersingular.
| $p$-rank: | $0$ |
| Slopes: | $[1/2, 1/2, 1/2, 1/2]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $400$ | $160000$ | $47059600$ | $16796160000$ | $6131071210000$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $438$ | $6860$ | $128878$ | $2476100$ | $47073318$ | $893871740$ | $16983041758$ | $322687697780$ | $6131076162198$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which all are hyperelliptic):
- $y^2=x^5+18$
- $y^2=2 x^5+17$
- $y^2=9 x^6+10 x^5+6 x^4+16 x^3+2 x^2+6 x+3$
- $y^2=16 x^6+4 x^4+4 x^2+16$
- $y^2=13 x^6+8 x^4+8 x^2+13$
- $y^2=16 x^6+8 x^4+16 x^2+14$
- $y^2=5 x^6+4 x^4+8 x^2+2$
- $y^2=2 x^6+17 x^5+15 x^4+11 x^2+8 x+16$
- $y^2=5 x^6+8 x^5+17 x^4+17 x^3+17 x^2+8 x+5$
- $y^2=10 x^6+16 x^5+15 x^4+15 x^3+15 x^2+16 x+10$
- $y^2=7 x^6+13 x^5+18 x^3+13 x+7$
- $y^2=14 x^6+7 x^5+17 x^3+7 x+14$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19^{2}}$.
Endomorphism algebra over $\F_{19}$| The isogeny class factors as 1.19.a 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-19}) \)$)$ |
| The base change of $A$ to $\F_{19^{2}}$ is 1.361.bm 2 and its endomorphism algebra is $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $19$ and $\infty$. |
Base change
This is a primitive isogeny class.