Invariants
Base field: | $\F_{2^{6}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 13 x + 64 x^{2} )^{2}$ |
$1 - 26 x + 297 x^{2} - 1664 x^{3} + 4096 x^{4}$ | |
Frobenius angles: | $\pm0.198106042756$, $\pm0.198106042756$ |
Angle rank: | $1$ (numerical) |
Jacobians: | $21$ |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2704$ | $16451136$ | $68876853136$ | $281693525577984$ | $1153062186256958224$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $39$ | $4015$ | $262743$ | $16790239$ | $1073872839$ | $68720346511$ | $4398049433271$ | $281474959033279$ | $18014398092657447$ | $1152921500319480175$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 21 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x)y=ax^5+(a^5+a+1)x^3+(a^4+a^2+a)x^2+(a^5+a^4+a^2+a+1)x$
- $y^2+(x^2+x)y=a^4x^5+(a^4+a^2+a)x^3+(a^5+a^4+a)x^2+(a^5+a^4+a^2)x$
- $y^2+(x^2+x)y=a^2x^5+(a^5+a^4+a^2)x^3+(a^5+a+1)x^2+(a^4+a+1)x$
- $y^2+(x^2+x)y=(a+1)x^5+(a+1)x^3+a^4x^2+a^4x$
- $y^2+(x^2+x)y=(a^2+1)x^5+(a^2+1)x^3+(a^5+a^4+a^2+a+1)x^2+(a^5+a^4+a^2+a+1)x$
- $y^2+(x^2+x)y=(a^4+1)x^5+(a^4+1)x^3+(a^4+a+1)x^2+(a^4+a+1)x$
- $y^2+(x^2+x)y=(a^5+a^4+a^2+a)x^5+(a^5+a^4+a^2+a)x^3+(a^5+a^4+a)x^2+(a^5+a^4+a)x$
- $y^2+(x^2+x)y=(a^5+a+1)x^5+(a^5+a+1)x^3+(a^5+a+1)x^2+(a^5+a+1)x$
- $y^2+(x^2+x)y=(a^4+a)x^5+(a^4+a)x^3+ax^2+ax$
- $y^2+(x^2+x)y=(a^5+a^4+a^2)x^5+(a^5+a^4+a^2)x^3+(a^5+a^4+a^2)x^2+(a^5+a^4+a^2)x$
- $y^2+(x^2+x)y=(a^5+a^4+a^2+a)x^5+x^3+(a+1)x^2+(a^5+a^4+a^2)x$
- $y^2+(x^2+x)y=(a^5+a^2+1)x^5+(a^5+a^2+1)x^3+(a^5+1)x^2+(a^5+1)x$
- $y^2+(x^2+x)y=(a^5+a^4+a+1)x^5+(a^5+a^4+a+1)x^3+a^2x^2+a^2x$
- $y^2+(x^2+x)y=(a^4+a^2+a)x^5+(a^4+a^2+a)x^3+(a^4+a^2+a)x^2+(a^4+a^2+a)x$
- $y^2+(x^2+x)y=(a^5+a^4+a+1)x^5+x^3+(a^5+a+1)x^2+(a^4+1)x$
- $y^2+(x^2+x)y=(a^5+a^2+a)x^5+(a^5+a^2+a)x^3+(a^5+a^2)x^2+(a^5+a^2)x$
- $y^2+(x^2+x)y=(a^4+a)x^5+x^3+(a^4+a^2+a)x^2+(a^2+1)x$
- $y^2+(x^2+x)y=(a^4+a^2+1)x^5+(a^4+a^2+1)x^3+(a^5+a^2+a+1)x^2+(a^5+a^2+a+1)x$
- $y^2+(x^2+x)y=(a^2+a+1)x^5+(a^2+a+1)x^3+(a^4+a^2)x^2+(a^4+a^2)x$
- $y^2+(x^2+x)y=(a^5+a^4+1)x^5+(a^5+a^4+1)x^3+(a^2+a)x^2+(a^2+a)x$
- $y^2+(x^2+x)y=a^5x^5+a^5x^3+(a^5+a^4)x^2+(a^5+a^4)x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{6}}$.
Endomorphism algebra over $\F_{2^{6}}$The isogeny class factors as 1.64.an 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-87}) \)$)$ |
Base change
This is a primitive isogeny class.