Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 + 31 x^{2} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.401848606134$, $\pm0.598151393866$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{7}, \sqrt{-69})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $14$ |
| Cyclic group of points: | yes |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $393$ | $154449$ | $47042100$ | $16921586889$ | $6131061313953$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $424$ | $6860$ | $129844$ | $2476100$ | $47038318$ | $893871740$ | $16983970084$ | $322687697780$ | $6131056370104$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which all are hyperelliptic):
- $y^2=8 x^6+17 x^5+7 x^4+7 x^3+9 x^2+x+10$
- $y^2=16 x^6+15 x^5+14 x^4+14 x^3+18 x^2+2 x+1$
- $y^2=17 x^6+12 x^5+5 x^3+4 x^2+18 x+4$
- $y^2=15 x^6+5 x^5+10 x^3+8 x^2+17 x+8$
- $y^2=10 x^6+13 x^5+9 x^4+18 x^3+8 x^2+8 x+4$
- $y^2=x^6+7 x^5+18 x^4+17 x^3+16 x^2+16 x+8$
- $y^2=14 x^6+7 x^5+18 x^4+9 x^3+6 x^2+3 x+14$
- $y^2=9 x^6+14 x^5+17 x^4+18 x^3+12 x^2+6 x+9$
- $y^2=8 x^6+14 x^4+11 x^3+6 x^2+15 x+11$
- $y^2=16 x^6+9 x^4+3 x^3+12 x^2+11 x+3$
- $y^2=12 x^6+15 x^5+8 x^4+8 x^3+11 x^2+2 x+18$
- $y^2=10 x^6+7 x^5+11 x^4+17 x^3+9 x^2+13 x+14$
- $y^2=x^6+14 x^5+3 x^4+15 x^3+18 x^2+7 x+9$
- $y^2=12 x^6+16 x^5+10 x^4+4 x^3+11 x^2+11 x+8$
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 \(\Q(\sqrt{7}, \sqrt{-69})\). |
| The base change of $A$ to $\F_{19^{2}}$ is 1.361.bf 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-483}) \)$)$ |
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 |
|---|---|---|
| 2.19.a_abf | $4$ | (not in LMFDB) |