Invariants
Base field: | $\F_{2^{5}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 13 x + 95 x^{2} - 416 x^{3} + 1024 x^{4}$ |
Frobenius angles: | $\pm0.163479829009$, $\pm0.410308350007$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.91225.1 |
Galois group: | $D_{4}$ |
Jacobians: | $30$ |
Isomorphism classes: | 30 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $691$ | $1070359$ | $1082324356$ | $1099589433931$ | $1125842719494361$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $20$ | $1046$ | $33029$ | $1048650$ | $33552730$ | $1073799407$ | $34360412848$ | $1099514142514$ | $35184366402893$ | $1125899815644686$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 30 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+(a^3+a^2)x+a^3+a^2)y=(a^4+a)x^5+(a^4+a)x^4+(a^4+a^3+a)x^3+ax+a^2$
- $y^2+(x^3+(a^4+a^3+a)x+a^4+a^3+a)y=(a^3+1)x^5+(a^3+1)x^4+(a+1)x^3+a^2x+a^4$
- $y^2+(x^3+(a^4+a^3+a^2+a)x+a^4+a^3+a^2+a)y=x^6+(a+1)x^5+(a+1)x^4+(a^4+a^3+a^2)x^3+(a^4+a+1)x^2+(a^4+a^3+a^2+1)x+a^3+1$
- $y^2+(x^3+(a+1)x+a+1)y=(a^3+a+1)x^5+(a^3+a+1)x^4+(a^2+1)x^3+a^4x+a^3+a^2+1$
- $y^2+(x^3+(a^4+a+1)x+a^4+a+1)y=x^6+x^5+x^4+(a^3+a)x^3+a^3x^2+(a^4+a^2)x+a^3+a^2+a+1$
- $y^2+(x^3+(a^4+a+1)x+a^4+a+1)y=x^6+(a^2+1)x^5+(a^2+1)x^4+(a^4+a^2+a+1)x^3+a^3x^2+(a^4+a^2+a)x+a^3+a+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+(a^3+a^2)x^5+(a^3+a^2)x^4+(a+1)x^3+(a^4+1)x^2+a^2x+a^3+a^2$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a^2+a+1)x^5+(a^3+a^2+a+1)x^4+(a^4+1)x^3+(a^3+a^2+1)x+a^4+a^3+a+1$
- $y^2+(x^3+(a^3+a^2+a)x+a^3+a^2+a)y=x^6+x^5+x^4+(a^4+a+1)x^3+(a^4+a^3+a^2+a)x^2+(a^4+a^3+1)x+a^3+1$
- $y^2+(x^3+a^3x+a^3)y=x^6+x^5+x^4+(a^3+a^2+a)x^3+(a^3+a)x^2+(a^4+a^3+a^2+1)x+a^4+a^3+a^2+a+1$
- $y^2+(x^3+(a+1)x+a+1)y=x^6+(a^4+a^2+a+1)x^5+(a^4+a^2+a+1)x^4+(a^4+a^3+a^2)x^3+(a^2+1)x^2+(a^4+a^3+a^2)x+a^2+1$
- $y^2+(x^3+a^3x+a^3)y=x^6+(a^4+1)x^5+(a^4+1)x^4+(a^4+a^3)x^3+(a^3+a)x^2+(a^4+a^3+1)x+a^3+a^2+a+1$
- $y^2+(x^3+(a^4+1)x+a^4+1)y=x^6+(a^2+a+1)x^5+(a^2+a+1)x^4+(a^4+a^3)x^3+(a^3+a^2)x^2+(a^4+a^3)x+a^3+a^2$
- $y^2+(x^3+(a^4+1)x+a^4+1)y=x^6+(a^4+a^3+a)x^5+(a^4+a^3+a)x^4+(a^2+1)x^3+(a^3+a^2)x^2+a^4x+a^4+a^3+a$
- $y^2+(x^3+(a^3+a)x+a^3+a)y=(a^2+a)x^5+(a^2+a)x^4+(a^3+a^2+a)x^3+(a^4+a^3+a+1)x+a$
- $y^2+(x^3+(a^4+1)x+a^4+1)y=(a^4+a^3+a^2+a+1)x^5+(a^4+a^3+a^2+a+1)x^4+(a^3+a^2)x^3+(a^4+a^3+a+1)x+a$
- $y^2+(x^3+(a^3+a^2+a)x+a^3+a^2+a)y=x^6+(a^4+a^3+a)x^5+(a^4+a^3+a)x^4+(a^4+a^2+1)x^3+(a^4+a^3+a^2+a)x^2+(a^4+a^2)x+a^4+a$
- $y^2+(x^3+(a^4+a^3+a^2+a)x+a^4+a^3+a^2+a)y=x^6+x^5+x^4+a^3x^3+(a^4+a+1)x^2+(a^2+a)x+a^3+a+1$
- $y^2+(x^3+(a+1)x+a+1)y=x^6+(a^4+1)x^5+(a^4+1)x^4+(a^4+a^3+a)x^3+(a^2+1)x^2+ax+a^4+1$
- $y^2+(x^3+(a^3+a)x+a^3+a)y=x^6+x^5+x^4+(a^4+a^3+a^2+a)x^3+(a^3+a^2+a)x^2+(a^4+a^2+a)x+a^4+a$
- $y^2+(x^3+(a^4+a^3+a)x+a^4+a^3+a)y=x^6+(a^4+a^3+a^2)x^5+(a^4+a^3+a^2)x^4+(a^4+a^2+1)x^3+(a+1)x^2+(a^4+a^2+1)x+a+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=x^6+(a^4+a^3)x^5+(a^4+a^3)x^4+(a^4+a^2+a+1)x^3+(a^4+1)x^2+(a^4+a^2+a+1)x+a^4+1$
- $y^2+(x^3+a^3x+a^3)y=(a^4+a^3+1)x^5+(a^4+a^3+1)x^4+(a^3+a)x^3+(a^3+a^2+1)x+a^4+a^3+a+1$
- $y^2+(x^3+(a^3+a)x+a^3+a)y=x^6+(a^3+a^2)x^5+(a^3+a^2)x^4+(a^2+a+1)x^3+(a^3+a^2+a)x^2+(a^2+a)x+a^4+a^3+a^2+a+1$
- $y^2+(x^3+(a^4+a^3+a)x+a^4+a^3+a)y=x^6+(a^2+1)x^5+(a^2+1)x^4+(a^3+a^2)x^3+(a+1)x^2+(a^4+a^3+a+1)x+a^2+1$
- $y^2+(x^3+(a^3+a^2)x+a^3+a^2)y=x^6+(a^4+a^2+1)x^5+(a^4+a^2+1)x^4+(a^2+a+1)x^3+(a^4+a^3+a)x^2+(a^2+a+1)x+a^4+a^3+a$
- $y^2+(x^3+(a^4+a+1)x+a^4+a+1)y=(a^4+a^2+a)x^5+(a^4+a^2+a)x^4+a^3x^3+a^4x+a^3+a^2+1$
- $y^2+(x^3+(a^3+a^2)x+a^3+a^2)y=x^6+(a+1)x^5+(a+1)x^4+(a^4+1)x^3+(a^4+a^3+a)x^2+(a^3+a^2+1)x+a+1$
- $y^2+(x^3+(a^4+a^3+a^2+a)x+a^4+a^3+a^2+a)y=(a^4+a^3+a^2+1)x^5+(a^4+a^3+a^2+1)x^4+(a^4+a+1)x^3+a^2x+a^4$
- $y^2+(x^3+(a^3+a^2+a)x+a^3+a^2+a)y=(a^4+a^2)x^5+(a^4+a^2)x^4+(a^4+a^3+a^2+a)x^3+ax+a^2$
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 4.0.91225.1. |
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_dr | $2$ | 2.1024.v_jx |