Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 7 x + 26 x^{2} - 63 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.122441590128$, $\pm0.422937410221$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.49708.1 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $38$ | $6764$ | $542336$ | $42423808$ | $3469878838$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $85$ | $744$ | $6465$ | $58763$ | $532594$ | $4791251$ | $43064513$ | $387425544$ | $3486785605$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+2)x^6+(a+2)x^5+(a+2)x^4+(2a+2)x^3+(2a+1)x^2+ax+a$
- $y^2=(a+2)x^6+2ax^5+x^4+(a+1)x^2+(2a+2)x+a+2$
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 4.0.49708.1. |
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.h_ba | $2$ | 2.81.d_abs |