Invariants
Base field: | $\F_{2^{5}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 11 x + 32 x^{2} )( 1 - 5 x + 32 x^{2} )$ |
$1 - 16 x + 119 x^{2} - 512 x^{3} + 1024 x^{4}$ | |
Frobenius angles: | $\pm0.0751336404065$, $\pm0.354289791955$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $25$ |
Isomorphism classes: | 100 |
This isogeny class is not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $616$ | $1029952$ | $1076331256$ | $1098806351104$ | $1125459241405576$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $17$ | $1007$ | $32849$ | $1047903$ | $33541297$ | $1073671247$ | $34359745297$ | $1099514101951$ | $35184388166513$ | $1125899945477807$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 25 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x+1)y=a^4x^5+x^4+a^4x^3+(a^4+a^3+a)x^2+(a^3+a+1)x+a^4+a^3+a$
- $y^2+(x^2+x+a+1)y=(a^4+a)x^5+x^4+(a^3+a+1)x^3+x^2+(a^2+a+1)x+a^3+a+1$
- $y^2+(x^2+x+a^3+a^2+a)y=(a^3+1)x^5+x^4+(a^4+a^2)x^3+x^2+(a^4+a^3+a^2+a+1)x+a^3+1$
- $y^2+(x^2+x+1)y=ax^5+x^4+ax^3+(a^4+1)x^2+(a^4+a)x+a^4+1$
- $y^2+(x^2+x+a^3+a^2)y=(a^3+a^2+a+1)x^5+x^4+(a^4+a)x^3+x^2+(a^4+a^2+a+1)x+a^4+a$
- $y^2+(x^2+x+1)y=(a^3+a^2+1)x^5+x^4+(a^3+a^2+1)x^3+(a+1)x^2+(a^3+a^2+a+1)x+a+1$
- $y^2+(x^2+x+a^2+1)y=(a^3+1)x^5+x^4+(a^3+a^2+a+1)x^3+x^2+(a^4+a^2+1)x+a^3+a^2+a+1$
- $y^2+(x^2+x)y=a^2x^5+(a^4+1)x^3+(a^2+1)x^2+a^4x$
- $y^2+(x^2+x)y=(a^3+a^2+1)x^5+(a^4+a^3+a)x^3+(a^3+a^2)x^2+(a^4+a^3+a+1)x$
- $y^2+(x^2+x+a^4+a^3+a^2+a)y=(a^3+a+1)x^5+x^4+(a^4+a^3+a^2+1)x^3+x^2+(a^4+a)x+a^3+a+1$
- $y^2+(x^2+x)y=a^3x^5+(a^4+a^3+a^2)x^3+(a^3+1)x^2+(a^4+a^3+a^2+1)x$
- $y^2+(x^2+x+1)y=a^2x^5+x^4+a^2x^3+(a^3+a^2)x^2+(a^3+1)x+a^3+a^2$
- $y^2+(x^2+x+a^4+a^3+a)y=(a^4+a^3+a^2+a+1)x^5+x^4+(a^3+1)x^3+x^2+(a^4+a^3)x+a^3+1$
- $y^2+(x^2+x+a^3+a)y=(a^4+a)x^5+x^4+(a^2+a)x^3+x^2+(a^3+a^2+a+1)x+a^4+a$
- $y^2+(x^2+x+1)y=(a^4+a^3+a+1)x^5+x^4+(a^4+a^3+a+1)x^3+(a^2+1)x^2+(a^4+a^3+a^2+a+1)x+a^2+1$
- $y^2+(x^2+x+a^4+1)y=(a^3+a+1)x^5+x^4+(a^4+a^3+a^2+a+1)x^3+x^2+(a^4+a^3+a^2)x+a^4+a^3+a^2+a+1$
- $y^2+(x^2+x)y=ax^5+(a^2+1)x^3+(a+1)x^2+a^2x$
- $y^2+(x^2+x)y=a^4x^5+(a^3+a^2)x^3+(a^4+1)x^2+(a^3+a^2+1)x$
- $y^2+(x^2+x)y=(a^4+a^2)x^5+(a^4+a^2+1)x^3+(a^4+a)x^2+(a^4+a+1)x$
- $y^2+(x^2+x+a^3)y=(a^4+a^3+a^2+a+1)x^5+x^4+(a^4+a^3+1)x^3+x^2+(a^3+a+1)x+a^4+a^3+a^2+a+1$
- $y^2+(x^2+x)y=ax^5+x^4+(a+1)x^3+(a^4+a^3+a)x^2+(a^4+a^3+a)x$
- $y^2+(x^2+x)y=(a^4+a^3+a^2+a+1)x^5+x^4+(a^2+a+1)x^3+(a^4+a^3+a^2+a+1)x^2+(a^2+a)x$
- $y^2+(x^2+x+a^4+a+1)y=(a^3+a^2+a+1)x^5+x^4+(a^4+a^2+a)x^3+x^2+(a^3+1)x+a^3+a^2+a+1$
- $y^2+(x^2+x)y=(a^3+a^2+a)x^5+x^4+(a^4+a^3)x^3+(a^4+a^3)x^2+(a^3+a^2+a+1)x$
- $y^2+(x^2+x)y=(a^3+a)x^5+x^4+(a^4+a^2+a+1)x^3+(a^4+a^2+a+1)x^2+(a^3+a+1)x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{5}}$.
Endomorphism algebra over $\F_{2^{5}}$The isogeny class factors as 1.32.al $\times$ 1.32.af and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is: |
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.ag_j | $2$ | 2.1024.as_agt |
2.32.g_j | $2$ | 2.1024.as_agt |
2.32.q_ep | $2$ | 2.1024.as_agt |