Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 5 x + 14 x^{2} - 45 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.100816880660$, $\pm0.537304361732$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.208444.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
Isomorphism classes: | 2 |
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: | $2$ |
Slopes: | $[0, 0, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $46$ | $6716$ | $496984$ | $41800384$ | $3503142926$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $85$ | $680$ | $6369$ | $59325$ | $533026$ | $4782405$ | $43049729$ | $387492440$ | $3486948325$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2x^5+ax^4+(2a+2)x^3+2ax^2+x+2a+1$
- $y^2=(a+1)x^5+ax^4+x^3+2ax^2+(2a+2)x+a+2$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{3^{2}}$.
Endomorphism algebra over $\F_{3^{2}}$The endomorphism algebra of this simple isogeny class is 4.0.208444.1. |
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 |
---|---|---|
2.9.f_o | $2$ | 2.81.d_ado |