Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 5 x^{2} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.228997657843$, $\pm0.771002342157$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-33}, \sqrt{43})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $18$ |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $357$ | $127449$ | $47051172$ | $17165978361$ | $6131063222277$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $352$ | $6860$ | $131716$ | $2476100$ | $47056462$ | $893871740$ | $16983112708$ | $322687697780$ | $6131060186752$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 18 curves (of which all are hyperelliptic):
- $y^2=6 x^6+7 x^5+8 x^4+14 x^3+3 x^2+14 x+4$
- $y^2=12 x^6+14 x^5+16 x^4+9 x^3+6 x^2+9 x+8$
- $y^2=15 x^6+7 x^5+13 x^4+7 x^3+17 x^2+14 x+11$
- $y^2=18 x^6+16 x^5+5 x^4+12 x^3+13 x^2+14$
- $y^2=17 x^6+13 x^5+10 x^4+5 x^3+7 x^2+9$
- $y^2=17 x^6+14 x^5+15 x^4+5 x^3+5 x^2+18 x+16$
- $y^2=12 x^6+x^5+10 x^3+18 x^2+x+12$
- $y^2=5 x^6+2 x^5+x^3+17 x^2+2 x+5$
- $y^2=6 x^6+17 x^5+18 x^4+18 x^3+8 x^2+14 x+13$
- $y^2=13 x^6+18 x^5+6 x^4+9 x^3+16 x^2+3 x+8$
- $y^2=18 x^6+9 x^5+4 x^4+3 x^3+15 x^2+12 x+16$
- $y^2=13 x^6+15 x^4+5 x^3+7 x^2+14 x+4$
- $y^2=7 x^6+11 x^4+10 x^3+14 x^2+9 x+8$
- $y^2=2 x^6+18 x^5+9 x^4+11 x^3+14 x^2+4 x+12$
- $y^2=4 x^6+17 x^5+18 x^4+3 x^3+9 x^2+8 x+5$
- $y^2=4 x^6+9 x^5+7 x^4+18 x^3+11 x^2+8 x+5$
- $y^2=8 x^6+18 x^5+14 x^4+17 x^3+3 x^2+16 x+10$
- $y^2=14 x^6+7 x^5+18 x^4+14 x^3+5 x^2+14 x+16$
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 \(\Q(\sqrt{-33}, \sqrt{43})\). |
| The base change of $A$ to $\F_{19^{2}}$ is 1.361.af 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-1419}) \)$)$ |
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_f | $4$ | (not in LMFDB) |