Invariants
Base field: | $\F_{2^{6}}$ |
Dimension: | $2$ |
L-polynomial: | $( 1 - 15 x + 64 x^{2} )( 1 - 13 x + 64 x^{2} )$ |
$1 - 28 x + 323 x^{2} - 1792 x^{3} + 4096 x^{4}$ | |
Frobenius angles: | $\pm0.113134082257$, $\pm0.198106042756$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $18$ |
Isomorphism classes: | 72 |
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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $2600$ | $16224000$ | $68668472600$ | $281563820928000$ | $1153006260223493000$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $37$ | $3959$ | $261949$ | $16782511$ | $1073820757$ | $68720190887$ | $4398051301933$ | $281474999945311$ | $18014398569066181$ | $1152921504430416599$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 18 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x+a^5+a^4+a^3+a+1)y=a^2x^5+a^2x^3+(a^5+a^4+a^3+a^2+a+1)x^2+a^2x+a^4+a^3+1$
- $y^2+(x^2+x+a^4+a^3+1)y=a^4x^5+(a^4+a^3+1)x^4+a^4x^3+(a^4+1)x^2+a^2$
- $y^2+(x^2+x+a^3+a^2)y=(a^5+a^4+a)x^5+(a^3+a^2)x^4+(a^5+a^4+a)x^3+(a^5+a^2+a+1)x^2+(a^5+a^4+a^2+a+1)x+1$
- $y^2+(x^2+x+a^3+a^2+a)y=(a^5+a^4+a^2+a+1)x^5+(a^5+a^4+a^2+a+1)x^3+(a^4+a^3+a+1)x^2+(a^4+1)x+a^5+a^4+a^2$
- $y^2+(x^2+x+a^3+a^2)y=a^4x^5+a^4x^3+(a^4+a^3+a^2+a)x^2+(a^4+a^3)x+a^5+a^4+a^3+a^2+a$
- $y^2+(x^2+x+a^5+a^3+1)y=(a^5+a^4+a)x^5+(a^5+a^3+1)x^4+(a^5+a^4+a)x^3+(a^4+a^2)x^2+(a^5+a^4+a^3+a^2+a)x+a^3$
- $y^2+(x^2+x+a^5+a^3+1)y=(a^2+a)x^5+(a^2+a)x^3+(a^3+a)x^2+(a^4+a^3+a^2+1)x+a^5+a^3+a^2+1$
- $y^2+(x^2+x+a^5+a^3+a)y=ax^5+(a^5+a^3+a)x^4+ax^3+(a^2+1)x^2+(a^5+a^3)x+a^5+a^3+a$
- $y^2+(x^2+x+a^3+a^2+a)y=(a^4+a+1)x^5+(a^3+a^2+a)x^4+(a^4+a+1)x^3+(a^5+a^2+a+1)x^2+(a^5+a^2+1)x+a^5+a^4+a^3+a^2+a$
- $y^2+(x^2+x+a^4+a^3+a)y=(a^4+a+1)x^5+(a^4+a+1)x^3+(a^4+a^3+a^2+1)x^2+(a^5+a^4+a^3+a^2+1)x+a^5+a^4+a^2+1$
- $y^2+(x^2+x+a^4+a^3+a+1)y=a^2x^5+a^2x^3+(a^3+a+1)x^2+(a^4+a^3+1)x+a^4+a^3$
- $y^2+(x^2+x+a^5+a^4+a^3+a^2+a)y=(a^5+a^4)x^5+(a^5+a^4)x^3+(a^3+a^2+1)x^2+(a^5+a^4+a^3+a^2)x+1$
- $y^2+(x^2+x+a^4+a^3+a+1)y=(a^5+a^4+a^2+a+1)x^5+(a^5+a^4+a^2+a+1)x^3+(a^5+a^4+a^3)x^2+(a^5+a^4+a^3+a^2+a+1)x+a^5+a^4+a^3+a^2+1$
- $y^2+(x^2+x+a^3+a^2+a)y=ax^5+ax^3+(a^3+a^2)x^2+(a^4+a)x+a^4+a+1$
- $y^2+(x^2+x+a^5+a^3+a)y=(a^5+1)x^5+(a^5+1)x^3+(a^5+a^4+a^3+a^2)x^2+(a^5+a^4+a^3+1)x+a^4+a^3+a^2$
- $y^2+(x^2+x+a^4+a^3+a)y=(a^5+a^2)x^5+(a^4+a^3+a)x^4+(a^5+a^2)x^3+(a^2+a)x^2+(a^3+a^2)x+a^3+a+1$
- $y^2+(x^2+x+a^5+a^3)y=(a^5+a^2+a+1)x^5+(a^5+a^2+a+1)x^3+(a^5+a^4+a^3+1)x^2+(a^4+a^3+a^2+a+1)x+1$
- $y^2+(x^2+x+a^3+a+1)y=(a^4+a^2)x^5+(a^4+a^2)x^3+(a^4+a^3+a^2+a+1)x^2+(a^2+a)x+a^5+a^4+a^2$
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.an 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.ac_acp | $2$ | (not in LMFDB) |
2.64.c_acp | $2$ | (not in LMFDB) |
2.64.bc_ml | $2$ | (not in LMFDB) |