Invariants
| Base field: | $\F_{19^{2}}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 7 x + 361 x^{2}$ |
| Frobenius angles: | $\pm0.441027178607$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-155}) \) |
| 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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $355$ | $130995$ | $47053120$ | $16983370755$ | $6131062298875$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $355$ | $130995$ | $47053120$ | $16983370755$ | $6131062298875$ | $2213314960769280$ | $799006687503978595$ | $288441413564614003395$ | $104127350297268876278080$ | $37589973457542547222879875$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 12 curves (of which 0 are hyperelliptic):
- $y^2=x^3+a^{310} x+a^{310}$
- $y^2=x^3+a^{103} x+a^{104}$
- $y^2=x^3+a^{158} x+a^{159}$
- $y^2=x^3+a^{86} x+a^{87}$
- $y^2=x^3+a^{239} x+a^{239}$
- $y^2=x^3+a^{221} x+a^{221}$
- $y^2=x^3+a^{121} x+a^{122}$
- $y^2=x^3+a^{272} x+a^{273}$
- $y^2=x^3+a^{130} x+a^{130}$
- $y^2=x^3+a^{48} x+a^{49}$
- $y^2=x^3+a^{92} x+a^{93}$
- $y^2=x^3+a^{156} x+a^{157}$
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 \(\Q(\sqrt{-155}) \). |
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.h | $2$ | (not in LMFDB) |