Invariants
| Base field: | $\F_{19^{2}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 30 x + 361 x^{2}$ |
| Frobenius angles: | $\pm0.210353590899$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-34}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $12$ |
| Isomorphism classes: | 12 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $332$ | $130144$ | $47051372$ | $16983792000$ | $6131071144652$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $332$ | $130144$ | $47051372$ | $16983792000$ | $6131071144652$ | $2213314983017824$ | $799006685937281132$ | $288441413549166528000$ | $104127350297301865021772$ | $37589973457534339022948704$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which 0 are hyperelliptic):
- $y^2=x^3+a^{114} x+a^{115}$
- $y^2=x^3+a^{211} x+a^{211}$
- $y^2=x^3+a^{330} x+a^{331}$
- $y^2=x^3+a^{34} x+a^{34}$
- $y^2=x^3+a^{176} x+a^{177}$
- $y^2=x^3+12 x+a^{301}$
- $y^2=x^3+a^{49} x+a^{50}$
- $y^2=x^3+a^{68} x+a^{69}$
- $y^2=x^3+a^{175} x+a^{176}$
- $y^2=x^3+a^{286} x+a^{286}$
- $y^2=x^3+a^{264} x+a^{265}$
- $y^2=x^3+a^{49} x+a^{49}$
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 \(\Q(\sqrt{-34}) \). |
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.be | $2$ | (not in LMFDB) |