Invariants
| Base field: | $\F_{311}$ |
| Dimension: | $1$ |
| L-polynomial: | $1 + 12 x + 311 x^{2}$ |
| Frobenius angles: | $\pm0.610504519859$ |
| Angle rank: | $1$ (numerical) |
| Number field: | \(\Q(\sqrt{-11}) \) |
| Galois group: | $C_2$ |
| Jacobians: | $20$ |
| Isomorphism classes: | 20 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $324$ | $97200$ | $30070764$ | $9354916800$ | $2909393387604$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $324$ | $97200$ | $30070764$ | $9354916800$ | $2909393387604$ | $904820267530800$ | $281399111678414844$ | $87515123964911251200$ | $27217203547656167696484$ | $8464550303313803493630000$ |
Jacobians and polarizations
This isogeny class is principally polarizable and contains the Jacobians of 20 curves (of which 0 are hyperelliptic):
- $y^2=x^3+205 x+205$
- $y^2=x^3+89 x+46$
- $y^2=x^3+80 x+258$
- $y^2=x^3+10 x+10$
- $y^2=x^3+169 x+169$
- $y^2=x^3+171 x+15$
- $y^2=x^3+160 x+205$
- $y^2=x^3+3 x+3$
- $y^2=x^3+261 x+72$
- $y^2=x^3+98 x+98$
- $y^2=x^3+212 x+212$
- $y^2=x^3+258 x+39$
- $y^2=x^3+75 x+75$
- $y^2=x^3+64 x+64$
- $y^2=x^3+73 x+73$
- $y^2=x^3+214 x+214$
- $y^2=x^3+118 x+118$
- $y^2=x^3+211 x+144$
- $y^2=x^3+9 x+99$
- $y^2=x^3+138 x+274$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{311}$.
Endomorphism algebra over $\F_{311}$| The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-11}) \). |
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 |
|---|---|---|
| 1.311.am | $2$ | (not in LMFDB) |