Invariants
| Base field: | $\F_{2^{5}}$ |
| Dimension: | $2$ |
| L-polynomial: | $1 - 15 x + 109 x^{2} - 480 x^{3} + 1024 x^{4}$ |
| Frobenius angles: | $\pm0.0910410022012$, $\pm0.380573396661$ |
| Angle rank: | $2$ (numerical) |
| Number field: | 4.0.28225.1 |
| Galois group: | $D_{4}$ |
| Jacobians: | $25$ |
| Isomorphism classes: | 35 |
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
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $639$ | $1040931$ | $1076679216$ | $1098474706611$ | $1125474689469339$ |
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $18$ | $1018$ | $32859$ | $1047586$ | $33541758$ | $1073710087$ | $34360076034$ | $1099515082978$ | $35184384856323$ | $1125899916316978$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 25 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+(a+1)x+a+1)y=x^6+(a^4+a^3+1)x^5+(a^4+a^3+1)x^4+(a^4+a^2+a+1)x^3+(a^2+1)x^2+a^3+a+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+(a^2+a)x^5+(a^2+a)x^4+(a^4+a^3)x^3+(a^4+1)x^2+a^3+a^2+a+1$
- $y^2+(x^3+(a^4+a^3+a^2+a)x+a^4+a^3+a^2+a)y=x^6+(a^4+a^3+1)x^3+(a^4+a+1)x^2+(a+1)x+a^4$
- $y^2+(x^3+(a^4+1)x+a^4+1)y=x^6+(a^4+a^2)x^5+(a^4+a^2)x^4+(a^2+a+1)x^3+(a^3+a^2)x^2+a^4+a^3+a^2+a+1$
- $y^2+(x^3+(a^3+a^2)x+a^3+a^2)y=x^6+(a^4+a^3+1)x^5+(a^4+a^3+1)x^4+(a^4+a^3+a+1)x^3+(a^4+a^3+a)x^2+a^4x+a^4+a^2+a+1$
- $y^2+(x^3+(a^3+a^2)x+a^3+a^2)y=x^6+(a^3+1)x^5+(a^3+1)x^4+(a^4+a^3+a+1)x^3+(a^4+a^3+a)x^2+(a^4+a)x+a^3+a+1$
- $y^2+(x^3+(a^4+a+1)x+a^4+a+1)y=x^6+(a^2+a)x^3+a^3x^2+(a^2+1)x+a^3+a^2+1$
- $y^2+(x^3+(a^3+a^2)x+a^3+a^2)y=x^6+(a^4+a^3+a^2+1)x^5+(a^4+a^3+a^2+1)x^4+(a^4+a^2+1)x^3+(a^4+a^3+a)x^2+a^4+a$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+(a^4+a^3+a^2+1)x^5+(a^4+a^3+a^2+1)x^4+a^4x^3+(a^4+1)x^2+ax+a^4+a^2+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+(a^4+a^3+a^2+a+1)x^5+(a^4+a^3+a^2+a+1)x^4+a^4x^3+(a^4+1)x^2+(a^3+a^2+a+1)x+a^4+a$
- $y^2+(x^3+(a^4+a^3+a)x+a^4+a^3+a)y=x^6+(a^2+a)x^5+(a^2+a)x^4+ax^3+(a+1)x^2+(a^3+a^2+1)x+a^4+a^3$
- $y^2+(x^3+(a^4+a^3+a)x+a^4+a^3+a)y=x^6+(a^3+a+1)x^5+(a^3+a+1)x^4+ax^3+(a+1)x^2+(a^3+1)x+a^3+a^2+a+1$
- $y^2+(x^3+(a^3+a^2)x+a^3+a^2)y=x^6+(a^4+a^3+a^2+a+1)x^5+(a^4+a^3+a^2+a+1)x^4+(a^3+a^2+1)x^3+(a^4+a^3+a)x^2+(a^4+a^3)x+a^4+a$
- $y^2+(x^3+a^3x+a^3)y=x^6+(a^4+a^2)x^3+(a^3+a)x^2+(a^4+1)x+a^4+a^3+a+1$
- $y^2+(x^3+(a+1)x+a+1)y=x^6+(a^3+1)x^5+(a^3+1)x^4+ax^3+(a^2+1)x^2+(a^4+a^2+1)x+a^3+a+1$
- $y^2+(x^3+(a^4+a^3+a)x+a^4+a^3+a)y=x^6+(a^4+a^2+a)x^5+(a^4+a^2+a)x^4+(a^4+a^3+a^2)x^3+(a+1)x^2+a^3+1$
- $y^2+(x^3+(a^3+a^2+a)x+a^3+a^2+a)y=x^6+(a^4+a^2+a)x^3+(a^4+a^3+a^2+a)x^2+(a^4+a^3+a)x+a^2$
- $y^2+(x^3+(a^4+1)x+a^4+1)y=x^6+(a^4+a)x^5+(a^4+a)x^4+(a^3+a^2+1)x^3+(a^3+a^2)x^2+(a^4+a^3+a^2+a+1)x+a^3+1$
- $y^2+(x^3+(a^4+1)x+a^4+1)y=x^6+(a^4+a^2+a)x^5+(a^4+a^2+a)x^4+(a^3+a^2+1)x^3+(a^3+a^2)x^2+a^2x+a^4+a^3+a^2$
- $y^2+(x^3+(a+1)x+a+1)y=x^6+(a^4+a^2)x^5+(a^4+a^2)x^4+a^2x^3+(a^2+1)x^2+(a^4+a^3+a+1)x+a^2+a+1$
- $y^2+(x^3+(a+1)x+a+1)y=x^6+(a^3+a^2+a+1)x^5+(a^3+a^2+a+1)x^4+a^2x^3+(a^2+1)x^2+(a^3+a+1)x+a^4+a^3+a^2+a+1$
- $y^2+(x^3+(a^4+1)x+a^4+1)y=x^6+(a^3+a^2+a+1)x^5+(a^3+a^2+a+1)x^4+a^4x^3+(a^3+a^2)x^2+(a^4+a^2+a+1)x+a^4+a^3+a^2+a+1$
- $y^2+(x^3+(a^4+a^3+a)x+a^4+a^3+a)y=x^6+(a^4+a)x^5+(a^4+a)x^4+(a^4+a^3+a+1)x^3+(a+1)x^2+(a^2+a+1)x+a^3+1$
- $y^2+(x^3+(a^3+a)x+a^3+a)y=x^6+(a^4+a^3+a^2+1)x^3+(a^3+a^2+a)x^2+(a^3+a^2)x+a$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+(a^3+a+1)x^5+(a^3+a+1)x^4+a^2x^3+(a^4+1)x^2+(a^4+a^3+a^2)x+a^3+a^2+a+1$
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 4.0.28225.1. |
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.p_ef | $2$ | 2.1024.ah_asd |