Invariants
Base field: | $\F_{331}$ |
Dimension: | $1$ |
L-polynomial: | $1 - 24 x + 331 x^{2}$ |
Frobenius angles: | $\pm0.270734319256$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-187}) \) |
Galois group: | $C_2$ |
Jacobians: | $8$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $308$ | $109648$ | $36274700$ | $12003824448$ | $3973197579428$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $308$ | $109648$ | $36274700$ | $12003824448$ | $3973197579428$ | $1315127785694800$ | $435307304962132988$ | $144086718334932350208$ | $47692703775667841729300$ | $15786284949779474868126928$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 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 \(\Q(\sqrt{-187}) \). |
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.y | $2$ | (not in LMFDB) |