Invariants
Base field: | $\F_{2^{5}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 13 x + 87 x^{2} - 416 x^{3} + 1024 x^{4}$ |
Frobenius angles: | $\pm0.0876507669829$, $\pm0.440213650551$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.6931001.1 |
Galois group: | $D_{4}$ |
Jacobians: | $20$ |
Isomorphism classes: | 20 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $683$ | $1052503$ | $1072055924$ | $1096974409259$ | $1125511333834393$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $20$ | $1030$ | $32717$ | $1046154$ | $33542850$ | $1073775295$ | $34360226480$ | $1099512723954$ | $35184369657989$ | $1125899948655150$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 20 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+x+1)y=(a^4+a)x^5+(a^4+a)x^4+(a^4+a^3+a^2)x^3+(a^4+a^3+a^2)x+a^3+a^2+a$
- $y^2+(x^3+x+1)y=(a^3+1)x^5+(a^3+1)x^4+(a^4+a^2+a+1)x^3+(a^4+a^2+a+1)x+a^4+a^3+a^2+a$
- $y^2+(x^3+x+1)y=(a^3+a+1)x^5+(a^3+a+1)x^4+(a^4+a^3)x^3+(a^4+a^3)x+a^4+a+1$
- $y^2+(x^3+(a^4+a^3+a)x+a^4+a^3+a)y=x^6+(a^2+1)x^5+(a^2+1)x^4+x^3+(a+1)x^2+a^3x+a^4+a^2+a$
- $y^2+(x^3+(a^4+a+1)x+a^4+a+1)y=x^6+(a^4+a^3+a)x^5+(a^4+a^3+a)x^4+(a^3+a)x^3+a^3x^2+(a^4+a^2+a)x+a^3+1$
- $y^2+(x^3+(a+1)x+a+1)y=x^6+x^5+x^4+(a^4+a^3+a^2)x^3+(a^2+1)x^2+(a^2+a+1)x+1$
- $y^2+(x^3+x+1)y=(a^3+a^2+a+1)x^5+(a^3+a^2+a+1)x^4+(a^2+a+1)x^3+(a^2+a+1)x+a^3$
- $y^2+(x^3+(a^4+1)x+a^4+1)y=x^6+(a^4+a^3+a)x^5+(a^4+a^3+a)x^4+x^3+(a^3+a^2)x^2+(a^4+a^3+a^2+a)x+a^4+a^2$
- $y^2+(x^3+(a+1)x+a+1)y=x^6+(a^4+1)x^5+(a^4+1)x^4+x^3+(a^2+1)x^2+(a^3+a)x+a^4+a^3+1$
- $y^2+(x^3+a^3x+a^3)y=x^6+(a+1)x^5+(a+1)x^4+(a^3+a^2+a)x^3+(a^3+a)x^2+(a^4+a^3+1)x+a^3+a+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+x^5+x^4+(a^4+a^2+a+1)x^3+(a^4+1)x^2+(a^4+a^2+1)x+1$
- $y^2+(x^3+x+1)y=(a^4+a^3+a^2+a+1)x^5+(a^4+a^3+a^2+a+1)x^4+(a^4+a^2+1)x^3+(a^4+a^2+1)x+a^3+a$
- $y^2+(x^3+(a^3+a^2)x+a^3+a^2)y=x^6+(a+1)x^5+(a+1)x^4+x^3+(a^4+a^3+a)x^2+(a^4+a+1)x+a^4+a^3+a^2+1$
- $y^2+(x^3+(a^4+a^3+a^2+a)x+a^4+a^3+a^2+a)y=x^6+(a^3+a^2)x^5+(a^3+a^2)x^4+a^3x^3+(a^4+a+1)x^2+(a^4+a^3+a^2+1)x+a^4+a$
- $y^2+(x^3+(a^4+a^3+a)x+a^4+a^3+a)y=x^6+x^5+x^4+(a^4+a^2+1)x^3+(a+1)x^2+(a^4+a^3)x+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+(a^3+a^2)x^5+(a^3+a^2)x^4+x^3+(a^4+1)x^2+(a^3+a^2+a)x+a^2+a$
- $y^2+(x^3+(a^3+a^2)x+a^3+a^2)y=x^6+x^5+x^4+(a^2+a+1)x^3+(a^4+a^3+a)x^2+(a^4+a^2+a+1)x+1$
- $y^2+(x^3+(a^3+a^2+a)x+a^3+a^2+a)y=x^6+(a^4+1)x^5+(a^4+1)x^4+(a^4+a+1)x^3+(a^4+a^3+a^2+a)x^2+(a^4+a^2)x+a^4+a^3+a^2+a+1$
- $y^2+(x^3+(a^4+1)x+a^4+1)y=x^6+x^5+x^4+(a^4+a^3)x^3+(a^3+a^2)x^2+(a^4+a^3+a^2)x+1$
- $y^2+(x^3+(a^3+a)x+a^3+a)y=x^6+(a^2+1)x^5+(a^2+1)x^4+(a^4+a^3+a^2+a)x^3+(a^3+a^2+a)x^2+(a^2+a)x+a^3+a^2+a+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{5}}$.
Endomorphism algebra over $\F_{2^{5}}$The endomorphism algebra of this simple isogeny class is 4.0.6931001.1. |
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 |
---|---|---|
2.32.n_dj | $2$ | 2.1024.f_abud |