Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 6 x + 19 x^{2} - 54 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.0763052093420$, $\pm0.490896535327$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.4672.2 |
Galois group: | $D_{4}$ |
Jacobians: | $3$ |
Isomorphism classes: | 3 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $41$ | $6601$ | $506432$ | $41408073$ | $3472388441$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $84$ | $694$ | $6308$ | $58804$ | $532686$ | $4784308$ | $43038404$ | $387432262$ | $3486984564$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 3 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=2ax^6+ax^5+2ax^3+2ax^2+ax+2a$
- $y^2=(2a+1)x^6+ax^5+(2a+2)x^4+(2a+1)x^3+2ax+2a+1$
- $y^2=(2a+1)x^6+2ax^5+(2a+1)x^4+2x^3+(2a+1)x^2+(2a+2)x+a$
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 4.0.4672.2. |
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.g_t | $2$ | 2.81.c_aev |