Invariants
Base field: | $\F_{2^{6}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 23 x + 245 x^{2} - 1472 x^{3} + 4096 x^{4}$ |
Frobenius angles: | $\pm0.0870708726221$, $\pm0.342563904315$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.13786305.2 |
Galois group: | $D_{4}$ |
Jacobians: | $24$ |
Isomorphism classes: | 24 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2847$ | $16617939$ | $68803699536$ | $281466424902195$ | $1152863693810239827$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $42$ | $4058$ | $262467$ | $16776706$ | $1073687982$ | $68718997271$ | $4398046603818$ | $281475019148386$ | $18014398977602043$ | $1152921507076730978$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 24 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+(a+1)x+a+1)y=(a^5+a^4+a^3+a^2)x^6+a^2x^5+a^2x^4+(a+1)x^3+(a^5+1)x^2+(a^4+a^3+a^2+1)x+a^5+a^4$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a^2+a+1)x^6+a^4x^5+a^4x^4+(a^2+1)x^3+(a^4+a)x^2+(a^5+a^4+a^3+a^2+1)x+a^5+a^2$
- $y^2+(x^3+(a^4+a^2+a)x+a^4+a^2+a)y=(a^4+a^3+a^2+a+1)x^6+(a^5+a^4+a^3+a^2)x^5+(a^5+a^4+a^3+a^2)x^4+ax^3+(a^5+a^3+a^2+a)x^2+(a^5+a^2+a+1)x+a^4+a^3$
- $y^2+(x^3+(a^4+1)x+a^4+1)y=(a^5+a^4+a^3+a^2)x^6+(a^5+a^4+a^2+a+1)x^5+(a^5+a^4+a^2+a+1)x^4+(a^4+1)x^3+(a^4+a^2)x^2+(a^3+a^2)x+a^2+a$
- $y^2+(x^3+(a^5+a^4+a^2+a)x+a^5+a^4+a^2+a)y=(a^5+a^4+a^3+a+1)x^6+(a^5+a)x^5+(a^5+a)x^4+(a^5+a^4+a^2)x^3+(a^5+a^3+1)x^2+(a^5+a^4+1)x+a^2+a$
- $y^2+(x^3+(a^5+a+1)x+a^5+a+1)y=(a^3+a^2+1)x^6+(a^3+a^2+1)x^5+(a^3+a^2+1)x^4+a^2x^3+(a^5+a^3)x^2+(a^5+a^2)x+a^5+a^4+a^3+a$
- $y^2+(x^3+(a^5+a^4+a^2+a)x+a^5+a^4+a^2+a)y=(a^4+a^3)x^6+(a^4+a+1)x^5+(a^4+a+1)x^4+(a^5+a^4+a^2+a)x^3+(a^5+a^4+1)x^2+(a^3+a+1)x+a$
- $y^2+(x^3+(a^4+a)x+a^4+a)y=(a^4+a^3+a^2+a)x^6+(a^5+a^4+a^2+1)x^5+(a^5+a^4+a^2+1)x^4+(a^4+a^2+a)x^3+(a^5+a^4+a^2+a+1)x^2+(a^2+a+1)x+a^3$
- $y^2+(x^3+(a^5+a^4+a^2)x+a^5+a^4+a^2)y=(a^4+a^3+a+1)x^6+(a^3+a)x^5+(a^3+a)x^4+a^4x^3+(a^5+a^4)x^2+(a^5+1)x+a+1$
- $y^2+(x^3+(a^4+a)x+a^4+a)y=(a^5+a^4+a^3+1)x^6+(a^5+a^4+a)x^5+(a^5+a^4+a)x^4+(a^4+a)x^3+(a^5+a^2+a+1)x^2+(a^4+a^3+a^2+a)x+a^5+a^2$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a^2+a+1)x^6+(a^5+a^4+a^2+1)x^5+(a^5+a^4+a^2+1)x^4+(a^4+a^2+a)x^3+(a^4+a)x^2+(a^5+a^2+1)x+a^5+a^4+a^3+a+1$
- $y^2+(x^3+(a^5+a^4+a+1)x+a^5+a^4+a+1)y=(a^5+a^3+a^2+a)x^6+(a^4+a^2+a+1)x^5+(a^4+a^2+a+1)x^4+(a^5+a+1)x^3+(a^5+a^3+1)x^2+(a^4+a^2+1)x+a^5+1$
- $y^2+(x^3+(a^5+a^2+a+1)x+a^5+a^2+a+1)y=a^3x^6+(a^4+a^2+a+1)x^5+(a^4+a^2+a+1)x^4+(a^2+1)x^3+(a^4+a^2+a+1)x^2+(a^5+a^4+a^2)x+a^5+a^2+1$
- $y^2+(x^3+(a^4+a^2+a)x+a^4+a^2+a)y=(a^5+a^4+a^3+a^2+a)x^6+(a^4+a^3+a^2+a+1)x^5+(a^4+a^3+a^2+a+1)x^4+(a^5+a^4+a^2+a+1)x^3+(a^2+a+1)x^2+(a^5+a^4)x+a^2$
- $y^2+(x^3+(a^5+1)x+a^5+1)y=(a^3+a^2+a)x^6+(a^5+a^4+a^2+1)x^5+(a^5+a^4+a^2+1)x^4+(a^5+a^4+a^2+a)x^3+(a^4+a^2+a+1)x^2+(a^5+a+1)x+a^4+a+1$
- $y^2+(x^3+(a^5+a^4+a+1)x+a^5+a^4+a+1)y=(a^4+a^3+a+1)x^6+ax^5+ax^4+(a^5+a^4+a+1)x^3+(a^5+a^3+a^2+1)x^2+(a^5+a^4+a^3)x+a^5+a^3$
- $y^2+(x^3+(a^5+a^4+a^2)x+a^5+a^4+a^2)y=(a^4+a^3+a^2+1)x^6+(a^4+a^3+a^2)x^5+(a^4+a^3+a^2)x^4+(a^5+a^4+a)x^3+(a^4+a^3+a^2+a+1)x^2+(a^4+a^2)x+a^3+a^2+a$
- $y^2+(x^3+(a^4+1)x+a^4+1)y=(a^5+a^4+a^3+1)x^6+(a^4+a^2+a+1)x^5+(a^4+a^2+a+1)x^4+(a^5+a+1)x^3+(a^4+a^2+a+1)x^2+a^5x+a^5+a^4+a^3+a^2$
- $y^2+(x^3+(a+1)x+a+1)y=(a^4+a^3+a^2+a+1)x^6+(a^5+a)x^5+(a^5+a)x^4+(a^5+a^4+a^2)x^3+(a^5+a^4+a^3)x^2+(a^5+a^2+a)x+a^5+a+1$
- $y^2+(x^3+(a^4+a^2)x+a^4+a^2)y=(a^4+a^3+a^2+a)x^6+(a^5+a^4+a^2+1)x^5+(a^5+a^4+a^2+1)x^4+(a+1)x^3+(a^5+a^3+a^2+a)x^2+(a^5+a+1)x+a^5+a^4+a^3+a^2+1$
- $y^2+(x^3+(a^5+a^2)x+a^5+a^2)y=(a^4+a^3+a^2+a+1)x^6+(a^5+a)x^5+(a^5+a)x^4+(a^4+1)x^3+a^4x^2+(a^4+a^2+a)x+a^4+a$
- $y^2+(x^3+(a^5+a+1)x+a^5+a+1)y=(a^3+a^2+a)x^6+(a^5+a^4+a^3+1)x^5+(a^5+a^4+a^3+1)x^4+(a^4+a+1)x^3+(a^5+a^3+a^2+a+1)x^2+(a^2+a)x+a^5+a^3+a$
- $y^2+(x^3+(a^2+a)x+a^2+a)y=a^3x^6+(a^5+a)x^5+(a^5+a)x^4+(a^5+a^4+a+1)x^3+(a^4+a^2+a)x^2+(a^4+a^2+a)x+a^5+1$
- $y^2+(x^3+(a^5+a^4)x+a^5+a^4)y=(a^5+a^3+a^2+a+1)x^6+(a^4+a^2+a+1)x^5+(a^4+a^2+a+1)x^4+(a^4+a)x^3+(a^3+a^2+a+1)x^2+(a^5+a^4+a^2)x+a^4+a^3+a^2+a+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{6}}$.
Endomorphism algebra over $\F_{2^{6}}$The endomorphism algebra of this simple isogeny class is 4.0.13786305.2. |
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 |
---|---|---|
2.64.x_jl | $2$ | (not in LMFDB) |