Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 29 x^{2} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.111823923823$, $\pm0.888176076177$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(i, \sqrt{67})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
This isogeny class is simple but not 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $333$ | $110889$ | $47052900$ | $16952821209$ | $6131070872253$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $304$ | $6860$ | $130084$ | $2476100$ | $47059918$ | $893871740$ | $16984056004$ | $322687697780$ | $6131075486704$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=8 x^6+x^5+17 x^4+17 x^3+17 x^2+5 x+17$
- $y^2=11 x^6+16 x^5+16 x^4+6 x^3+3 x^2+6 x+9$
- $y^2=3 x^6+13 x^5+13 x^4+12 x^3+6 x^2+12 x+18$
- $y^2=2 x^6+13 x^5+14 x^4+8 x^3+17 x^2+x+9$
- $y^2=4 x^6+7 x^5+9 x^4+16 x^3+15 x^2+2 x+18$
- $y^2=16 x^6+9 x^5+5 x^4+15 x^3+2 x^2+11$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19^{2}}$.
Endomorphism algebra over $\F_{19}$| The endomorphism algebra of this simple isogeny class is \(\Q(i, \sqrt{67})\). |
| The base change of $A$ to $\F_{19^{2}}$ is 1.361.abd 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-67}) \)$)$ |
Base change
This is a primitive isogeny class.