Invariants
Base field: | $\F_{3^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 6 x + 20 x^{2} - 54 x^{3} + 81 x^{4}$ |
Frobenius angles: | $\pm0.109926884584$, $\pm0.481195587521$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.116032.1 |
Galois group: | $D_{4}$ |
Jacobians: | $8$ |
Isomorphism classes: | 8 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $42$ | $6804$ | $519498$ | $41830992$ | $3482994522$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $4$ | $86$ | $712$ | $6374$ | $58984$ | $533510$ | $4788004$ | $43047230$ | $387439876$ | $3486994886$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 8 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2=(a+2)x^6+2ax^5+(a+1)x^4+2x^3+(2a+1)x^2+(a+2)x+a+2$
- $y^2=2x^6+(2a+2)x^5+x^4+(2a+2)x^3+(a+2)x^2+x+a+2$
- $y^2=(2a+1)x^6+2x^5+x^4+x^3+(2a+1)x+a+1$
- $y^2=(a+2)x^6+x^5+2x^3+(a+2)x^2+(2a+1)x+a+2$
- $y^2=(2a+1)x^6+x^4+(2a+2)x^3+(a+2)x^2+x+a+2$
- $y^2=ax^6+2ax^5+(2a+2)x^4+ax^3+x^2+2x+a+1$
- $y^2=(a+2)x^6+(a+2)x^5+ax^4+(a+2)x^3+x^2+ax+a+2$
- $y^2=(2a+1)x^6+(a+2)x^5+x^4+x^3+(2a+1)x^2+ax+2a$
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 4.0.116032.1. |
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.g_u | $2$ | 2.81.e_adi |