Invariants
| Base field: | $\F_{337}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 13 x + 337 x^{2}$ |
| Frobenius angles: | $\pm0.615205115990$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-131}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $15$ |
| Isomorphism classes: | 15 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $351$ | $114075$ | $38261808$ | $12897889875$ | $4346602336791$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $351$ | $114075$ | $38261808$ | $12897889875$ | $4346602336791$ | $1464803578929600$ | $493638819878269143$ | $166356282594537403875$ | $56062067225873294894256$ | $18892916655130011955192875$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 15 curves (of which 0 are hyperelliptic):
- $y^2=x^3+165 x+165$
- $y^2=x^3+57 x+285$
- $y^2=x^3+110 x+213$
- $y^2=x^3+293 x+293$
- $y^2=x^3+65 x+65$
- $y^2=x^3+98 x+98$
- $y^2=x^3+162 x+162$
- $y^2=x^3+298 x+298$
- $y^2=x^3+252 x+249$
- $y^2=x^3+232 x+232$
- $y^2=x^3+90 x+90$
- $y^2=x^3+13 x+13$
- $y^2=x^3+237 x+237$
- $y^2=x^3+133 x+133$
- $y^2=x^3+218 x+218$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{337}$.
Endomorphism algebra over $\F_{337}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-131}) \). |
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 |
|---|---|---|
| 1.337.an | $2$ | (not in LMFDB) |