Invariants
Base field: | $\F_{2^{5}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 17 x + 131 x^{2} - 544 x^{3} + 1024 x^{4}$ |
Frobenius angles: | $\pm0.0971087340683$, $\pm0.315093300853$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.455553.1 |
Galois group: | $D_{4}$ |
Jacobians: | $10$ |
Isomorphism classes: | 10 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $595$ | $1021615$ | $1078185220$ | $1100243598475$ | $1125791024138725$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $16$ | $998$ | $32905$ | $1049274$ | $33551186$ | $1073696951$ | $34359640772$ | $1099513367026$ | $35184393354745$ | $1125900033379718$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+a^3x+a^3)y=(a^4+a^3+a)x^5+(a^4+a^3+a)x^4+(a^4+a^3+a+1)x^3+(a^4+a^2+a+1)x+a^4+a^3+1$
- $y^2+(x^3+(a^4+a^3+a)x+a^4+a^3+a)y=x^6+(a^4+a^2+a+1)x^5+(a^4+a^2+a+1)x^4+x^3+(a+1)x^2+(a^4+a^2+a)x+a^2+1$
- $y^2+(x^3+(a^4+a^3+a^2+a)x+a^4+a^3+a^2+a)y=(a^4+1)x^5+(a^4+1)x^4+a^4x^3+(a^4+a^2+1)x+a^4+a^3+a^2+1$
- $y^2+(x^3+(a^3+a)x+a^3+a)y=(a+1)x^5+(a+1)x^4+ax^3+(a^4+a^3)x+a^2+a$
- $y^2+(x^3+(a+1)x+a+1)y=x^6+(a^4+a^3)x^5+(a^4+a^3)x^4+x^3+(a^2+1)x^2+(a^4+a^3+1)x+a^4+1$
- $y^2+(x^3+(a^4+a+1)x+a^4+a+1)y=(a^3+a^2)x^5+(a^3+a^2)x^4+(a^3+a^2+1)x^3+(a^4+a^3+a^2)x+a^4+a^2+a$
- $y^2+(x^3+(a^3+a^2)x+a^3+a^2)y=x^6+(a^4+a^3+a^2)x^5+(a^4+a^3+a^2)x^4+x^3+(a^4+a^3+a)x^2+(a^4+a^3+a^2+1)x+a+1$
- $y^2+(x^3+(a^3+a^2+a)x+a^3+a^2+a)y=(a^2+1)x^5+(a^2+1)x^4+a^2x^3+(a^2+a+1)x+a^4+a^2$
- $y^2+(x^3+(a^4+1)x+a^4+1)y=x^6+(a^4+a^2+1)x^5+(a^4+a^2+1)x^4+x^3+(a^3+a^2)x^2+(a^4+a^2)x+a^4+a^3+a$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+(a^2+a+1)x^5+(a^2+a+1)x^4+x^3+(a^4+1)x^2+(a^2+a)x+a^3+a^2$
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 4.0.455553.1. |
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.r_fb | $2$ | 2.1024.abb_bbl |