Invariants
Base field: | $\F_{2^{5}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 13 x + 85 x^{2} - 416 x^{3} + 1024 x^{4}$ |
Frobenius angles: | $\pm0.0605291260618$, $\pm0.446568083608$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.4111025.1 |
Galois group: | $D_{4}$ |
Jacobians: | $10$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $681$ | $1048059$ | $1069494156$ | $1096279146531$ | $1125384870777261$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $20$ | $1026$ | $32639$ | $1045490$ | $33539080$ | $1073749527$ | $34359992428$ | $1099511050114$ | $35184361909703$ | $1125899921416986$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 10 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+(a^3+a)x+a^3+a)y=x^6+(a^4+a^2+a+1)x^5+(a^4+a^2+a+1)x^4+(a^4+a^3)x^3+(a^3+a^2+a)x^2+(a^4+a^3+a^2+1)x+a^4+1$
- $y^2+(x^3+(a^3+a^2+a)x+a^3+a^2+a)y=x^6+(a^4+a^3)x^5+(a^4+a^3)x^4+(a^2+a+1)x^3+(a^4+a^3+a^2+a)x^2+(a^4+a^2+a)x+a^3+a^2$
- $y^2+(x^3+(a+1)x+a+1)y=x^6+(a^3+a^2+1)x^5+(a^3+a^2+1)x^4+(a^2+a)x^3+(a^2+1)x^2+x+a^4+a+1$
- $y^2+(x^3+a^3x+a^3)y=x^6+(a^4+a^3+a^2)x^5+(a^4+a^3+a^2)x^4+(a^4+a^2+a+1)x^3+(a^3+a)x^2+(a^4+a^2)x+a^2+1$
- $y^2+(x^3+(a^4+a^3+a)x+a^4+a^3+a)y=x^6+a^4x^5+a^4x^4+(a^4+a^3+1)x^3+(a+1)x^2+x+a^4+a^3+a^2+a$
- $y^2+(x^3+(a^4+a+1)x+a^4+a+1)y=x^6+(a^4+a^2+1)x^5+(a^4+a^2+1)x^4+(a^4+a^3+a^2)x^3+a^3x^2+(a^2+a)x+a+1$
- $y^2+(x^3+(a^3+a^2)x+a^3+a^2)y=x^6+a^2x^5+a^2x^4+(a^4+a^2+a)x^3+(a^4+a^3+a)x^2+x+a^3+a^2+a$
- $y^2+(x^3+(a^4+a^3+a^2+a)x+a^4+a^3+a^2+a)y=x^6+(a^2+a+1)x^5+(a^2+a+1)x^4+(a^4+a^2+1)x^3+(a^4+a+1)x^2+(a^4+a^3+1)x+a^4+a^3+a$
- $y^2+(x^3+(a^4+1)x+a^4+1)y=x^6+ax^5+ax^4+(a^4+a^3+a^2+1)x^3+(a^3+a^2)x^2+x+a^3+a$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+(a^4+a^3+a+1)x^5+(a^4+a^3+a+1)x^4+(a^4+a^2)x^3+(a^4+1)x^2+x+a^3$
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 4.0.4111025.1. |
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_dh | $2$ | 2.1024.b_achj |