Invariants
Base field: | $\F_{331}$ |
Dimension: | $1$ |
L-polynomial: | $1 - 9 x + 331 x^{2}$ |
Frobenius angles: | $\pm0.420442761943$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-1243}) \) |
Galois group: | $C_2$ |
Jacobians: | $4$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $323$ | $110143$ | $36272900$ | $12003494283$ | $3973192027853$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $323$ | $110143$ | $36272900$ | $12003494283$ | $3973192027853$ | $1315127818483600$ | $435307307509263743$ | $144086718365732452563$ | $47692703775414252475100$ | $15786284949768293868390703$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{331}$.
Endomorphism algebra over $\F_{331}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-1243}) \). |
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 |
---|---|---|
1.331.j | $2$ | (not in LMFDB) |