Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 7 x + 27 x^{2} - 63 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.154979380638$, $\pm0.408713257520$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.4901.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $39$ | $6981$ | $558207$ | $43009941$ | $3480111024$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $87$ | $765$ | $6555$ | $58938$ | $532647$ | $4791153$ | $43064979$ | $387412335$ | $3486652302$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(2a+1)x^6+x^4+(a+2)x^3+(2a+1)x+2a+1$
- $y^2=ax^6+x^4+2ax^3+ax+a$
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 4.0.4901.1. |
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_bb | $2$ | 2.81.f_j |