Invariants
Base field: | $\F_{2^{3}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 5 x + 8 x^{2} )( 1 + 3 x + 8 x^{2} )$ |
$1 - 2 x + x^{2} - 16 x^{3} + 64 x^{4}$ | |
Frobenius angles: | $\pm0.154919815756$, $\pm0.677932000632$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $12$ |
Isomorphism classes: | 102 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $48$ | $4032$ | $237744$ | $17305344$ | $1086883248$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $7$ | $63$ | $463$ | $4223$ | $33167$ | $262143$ | $2101967$ | $16785151$ | $134203279$ | $1073782143$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 12 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x+1)y=a^2x^5+(a^2+a+1)x^3+a^2x+a+1$
- $y^2+(x^2+x+1)y=(a^2+a)x^5+(a+1)x^3+(a^2+a)x+a^2+1$
- $y^2+(x^2+x)y=x^5+x^4+ax^3+a^2x^2+(a^2+a)x$
- $y^2+(x^2+x+a+1)y=ax^5+x^3+(a^2+a+1)x+a^2+1$
- $y^2+(x^2+x)y=a^2x^5+a^2x^3+(a^2+a)x^2+(a^2+a)x$
- $y^2+(x^2+x+1)y=ax^5+(a^2+1)x^3+ax+a^2+a+1$
- $y^2+(x^2+x)y=x^5+x^4+(a^2+a)x^3+ax^2+a^2x$
- $y^2+(x^2+x+a^2+a+1)y=(a^2+a)x^5+x^3+(a^2+1)x+a+1$
- $y^2+(x^2+x)y=ax^5+ax^3+a^2x^2+a^2x$
- $y^2+(x^2+x)y=x^5+x^4+a^2x^3+(a^2+a)x^2+ax$
- $y^2+(x^2+x+a^2+1)y=a^2x^5+x^3+(a+1)x+a^2+a+1$
- $y^2+(x^2+x)y=(a^2+a)x^5+(a^2+a)x^3+ax^2+ax$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{3}}$.
Endomorphism algebra over $\F_{2^{3}}$The isogeny class factors as 1.8.af $\times$ 1.8.d 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 |
---|---|---|
2.8.ai_bf | $2$ | 2.64.ac_cn |
2.8.c_b | $2$ | 2.64.ac_cn |
2.8.i_bf | $2$ | 2.64.ac_cn |