Invariants
| Base field: | $\F_{349}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 - 16 x + 349 x^{2}$ |
| Frobenius angles: | $\pm0.359137247884$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-285}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $16$ |
| Isomorphism classes: | 16 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $334$ | $122244$ | $42521206$ | $14835531840$ | $5177580131614$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $334$ | $122244$ | $42521206$ | $14835531840$ | $5177580131614$ | $1806976662928164$ | $630634881661441606$ | $220091573702883989760$ | $76811959213176636401134$ | $26807373765251506824104004$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 16 curves (of which 0 are hyperelliptic):
- $y^2=x^3+36 x+72$
- $y^2=x^3+320 x+291$
- $y^2=x^3+211 x+211$
- $y^2=x^3+38 x+38$
- $y^2=x^3+194 x+194$
- $y^2=x^3+152 x+152$
- $y^2=x^3+5 x+5$
- $y^2=x^3+176 x+176$
- $y^2=x^3+310 x+310$
- $y^2=x^3+257 x+257$
- $y^2=x^3+188 x+27$
- $y^2=x^3+282 x+215$
- $y^2=x^3+85 x+85$
- $y^2=x^3+295 x+295$
- $y^2=x^3+35 x+35$
- $y^2=x^3+339 x+339$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{349}$.
Endomorphism algebra over $\F_{349}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-285}) \). |
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 |
|---|---|---|
| 1.349.q | $2$ | (not in LMFDB) |