Invariants
| Base field: | $\F_{19^{2}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 12 x + 361 x^{2}$ |
| Frobenius angles: | $\pm0.397730665719$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-13}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $14$ |
| Isomorphism classes: | 14 |
This isogeny class is simple and geometrically simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $1$ |
| Slopes: | $[0, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $350$ | $130900$ | $47057150$ | $16983489600$ | $6131061308750$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $350$ | $130900$ | $47057150$ | $16983489600$ | $6131061308750$ | $2213314886190100$ | $799006687174979150$ | $288441413596194566400$ | $104127350297751576012350$ | $37589973457533727210172500$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 14 curves (of which 0 are hyperelliptic):
- $y^2=x^3+a^{155} x+a^{156}$
- $y^2=x^3+a^{50} x+a^{50}$
- $y^2=x^3+a^{254} x+a^{254}$
- $y^2=x^3+a^{146} x+a^{146}$
- $y^2=x^3+a^{91} x+a^{91}$
- $y^2=x^3+a^{314} x+a^{314}$
- $y^2=x^3+a^{29} x+a^{30}$
- $y^2=x^3+a^{230} x+a^{230}$
- $y^2=x^3+a^{149} x+a^{149}$
- $y^2=x^3+a^{308} x+a^{309}$
- $y^2=x^3+a^{206} x+a^{206}$
- $y^2=x^3+a^{311} x+a^{311}$
- $y^2=x^3+a^{289} x+a^{289}$
- $y^2=x^3+a^{56} x+a^{57}$
where $a$ is a root of the Conway polynomial.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{19^{2}}$.
Endomorphism algebra over $\F_{19^{2}}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-13}) \). |
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 |
|---|---|---|
| 1.361.m | $2$ | (not in LMFDB) |