Invariants
Base field: | $\F_{2^{6}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 15 x + 64 x^{2} )( 1 - 11 x + 64 x^{2} )$ |
$1 - 26 x + 293 x^{2} - 1664 x^{3} + 4096 x^{4}$ | |
Frobenius angles: | $\pm0.113134082257$, $\pm0.258708130235$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $48$ |
Isomorphism classes: | 240 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2700$ | $16416000$ | $68794587900$ | $281591199360000$ | $1152978430634923500$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $39$ | $4007$ | $262431$ | $16784143$ | $1073794839$ | $68719670327$ | $4398046364751$ | $281474976029983$ | $18014398639654599$ | $1152921507154294727$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 48 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x)y=(a^5+a+1)x^5+(a^5+a^4+a^3+a+1)x^4+(a^5+a+1)x^3+(a^4+1)x^2+(a^5+a^3+a)x$
- $y^2+(x^2+x)y=(a^5+a^4+a^2)x^5+(a^3+a^2)x^4+(a^5+a^4+a^2)x^3+(a^5+a)x^2+(a^5+a^3+a^2+a)x$
- $y^2+(x^2+x+a^5+a^3)y=(a^5+a^2+a+1)x^5+(a^5+a^3+a^2+1)x^4+(a^3+1)x^3+(a^5+a^3+a^2+1)x^2+(a^5+a^4+a^3+a^2+a+1)x+a^5+a$
- $y^2+(x^2+x)y=(a^4+a^2+a)x^5+(a^4+a^3+a+1)x^4+(a^4+a^2+a)x^3+(a^5+a^2+1)x^2+(a^5+a^4+a^3+a^2+a)x$
- $y^2+(x^2+x)y=(a^5+a^2)x^5+(a^4+a^3+1)x^4+(a^5+a^2)x^3+(a^4+a^2+1)x^2+(a^3+a^2)x$
- $y^2+(x^2+x+a^5+a^3+a)y=(a^5+a^2)x^5+(a^4+a^3+a^2+a+1)x^4+(a^4+a^3+a)x^3+(a^4+a^3+a^2+a+1)x^2+a^3x+a^4$
- $y^2+(x^2+x)y=(a^5+a+1)x^5+(a^5+a^3+a^2+a)x^4+(a^5+a+1)x^3+(a^5+a^2+a+1)x^2+(a^3+1)x$
- $y^2+(x^2+x+a^5+a^4+a^3+a)y=(a^4+1)x^5+(a^5+a^4+a^3+a^2)x^4+(a^4+a^3+a^2+1)x^3+(a^5+a^4+a^3+a^2)x^2+(a^3+a+1)x+a^5+a^2+a$
- $y^2+(x^2+x+a^5+a^4+a^3+a+1)y=(a^5+a^4+a+1)x^5+(a^3+a^2+a)x^4+(a^3+a+1)x^3+(a^3+a^2+a)x^2+(a^4+a^3+a^2+1)x+a^5+a^3$
- $y^2+(x^2+x)y=(a^5+1)x^5+(a^5+a^3+a^2+1)x^4+(a^5+1)x^3+(a^5+a^2+a)x^2+(a^3+a+1)x$
- $y^2+(x^2+x+a^5+a^3+a^2+a)y=(a^5+1)x^5+(a^5+a^4+a^3+a+1)x^4+(a^5+a^3)x^3+(a^5+a^4+a^3+a+1)x^2+(a^4+a^3+a+1)x+a^4$
- $y^2+(x^2+x)y=(a^5+a^2+a)x^5+a^3x^4+(a^5+a^2+a)x^3+(a^5+a^4+a+1)x^2+(a^5+a^4+a^3+a+1)x$
- $y^2+(x^2+x+a^5+a^3+a^2+a+1)y=ax^5+(a^3+a^2)x^4+(a^4+a^3+a^2+a+1)x^3+(a^3+a^2)x^2+(a^5+1)x+a^5+a^3+a$
- $y^2+(x^2+x)y=(a^5+a^4+a^2)x^5+(a^5+a^3+a+1)x^4+(a^5+a^4+a^2)x^3+(a^5+a^4+1)x^2+(a^4+a^3+a)x$
- $y^2+(x^2+x)y=(a^4+a^2)x^5+(a^4+a^3+a^2+a)x^4+(a^4+a^2)x^3+(a+1)x^2+(a^4+a^3+a^2+1)x$
- $y^2+(x^2+x+a^4+a^3+1)y=(a^5+a^4+a^2+a)x^5+(a^5+a^4+a^3+1)x^4+(a^5+a^4+a^3+a^2+1)x^3+(a^5+a^4+a^3+1)x^2+(a^4+a^3+a^2+a)x+a^5+a^4+a^3+a^2+a+1$
- $y^2+(x^2+x+a^4+a^3)y=(a+1)x^5+(a^3+a+1)x^4+(a^4+a^3+a^2+a)x^3+(a^3+a+1)x^2+(a^5+a^4+a^3+a^2+1)x+a^5+a^3+a^2$
- $y^2+(x^2+x)y=(a^2+a+1)x^5+(a^3+a^2+a)x^4+(a^2+a+1)x^3+(a^4+a^2+a+1)x^2+(a^4+a^3+1)x$
- $y^2+(x^2+x)y=(a^5+a^4)x^5+a^3x^4+(a^5+a^4)x^3+(a^4+a^2+a)x^2+(a^4+a^3+a^2+a)x$
- $y^2+(x^2+x+a^5+a^4+a^3+a^2+a)y=(a^5+a^4)x^5+(a^4+a^3+a^2+1)x^4+(a^5+a^3+a)x^3+(a^4+a^3+a^2+1)x^2+(a^5+a^3+1)x+a^4+a^3+a$
- and 28 more
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.ap $\times$ 1.64.al 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 |
---|---|---|
2.64.ae_abl | $2$ | (not in LMFDB) |
2.64.e_abl | $2$ | (not in LMFDB) |
2.64.ba_lh | $2$ | (not in LMFDB) |