Invariants
| Base field: | $\F_{19}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 27 x^{2} + 361 x^{4}$ |
| Frobenius angles: | $\pm0.124228461180$, $\pm0.875771538820$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-11}, \sqrt{65})\) |
| Galois group: | $C_2^2$ |
| Jacobians: | $6$ |
| Isomorphism classes: | 16 |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $335$ | $112225$ | $47055440$ | $16981999225$ | $6131069843375$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $20$ | $308$ | $6860$ | $130308$ | $2476100$ | $47064998$ | $893871740$ | $16984084228$ | $322687697780$ | $6131073428948$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 6 curves (of which all are hyperelliptic):
- $y^2=2 x^6+6 x^5+18 x^4+5 x^3+5 x^2+17 x+16$
- $y^2=16 x^6+10 x^5+5 x^4+14 x^3+14 x+16$
- $y^2=13 x^6+x^5+10 x^4+9 x^3+9 x+13$
- $y^2=16 x^6+15 x^5+6 x^4+4 x^3+7 x^2+5$
- $y^2=13 x^6+11 x^5+12 x^4+8 x^3+14 x^2+10$
- $y^2=4 x^6+18 x^4+2 x^3+7 x^2+15$
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 \(\Q(\sqrt{-11}, \sqrt{65})\). |
| The base change of $A$ to $\F_{19^{2}}$ is 1.361.abb 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-715}) \)$)$ |
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_bb | $4$ | (not in LMFDB) |