Invariants
Base field: | $\F_{3^{5}}$ |
Dimension: | $1$ |
L-polynomial: | $1 - 22 x + 243 x^{2}$ |
Frobenius angles: | $\pm0.250654960462$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-122}) \) |
Galois group: | $C_2$ |
Jacobians: | $10$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $222$ | $59052$ | $14354298$ | $3486902496$ | $847289897742$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $222$ | $59052$ | $14354298$ | $3486902496$ | $847289897742$ | $205891131740364$ | $50031544778149002$ | $12157665452084304768$ | $2954312706475402691454$ | $717897987691887454252332$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{5}}$.
Endomorphism algebra over $\F_{3^{5}}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-122}) \). |
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 |
---|---|---|
1.243.w | $2$ | (not in LMFDB) |