Invariants
Base field: | $\F_{2^{4}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 7 x + 33 x^{2} - 112 x^{3} + 256 x^{4}$ |
Frobenius angles: | $\pm0.172472086823$, $\pm0.494194579844$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{-3}, \sqrt{5})\) |
Galois group: | $C_2^2$ |
Jacobians: | $29$ |
Isomorphism classes: | 23 |
This isogeny class is simple but not geometrically simple, not 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $171$ | $69939$ | $16842816$ | $4280336739$ | $1101267647451$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $10$ | $274$ | $4111$ | $65314$ | $1050250$ | $16793503$ | $268465690$ | $4294885954$ | $68719305391$ | $1099512329554$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 29 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^3+(a^2+1)x+a^2+1)y=a^3x^6+(a^2+1)x^5+(a^2+1)x^4+(a^2+1)x^3+(a+1)x^2+(a^3+a)x+a^3+a+1$
- $y^2+(x^3+a^2x+a^2)y=a^3x^6+x^5+x^4+(a+1)x^3+(a^3+a+1)x^2+(a^3+a+1)x+a^2$
- $y^2+(x^3+a^3+a+1)y=(a^3+a^2+a+1)x^6+(a^3+a+1)x^4+x^3+(a^3+1)x+a^2+1$
- $y^2+(x^3+a^3+1)y=(a^3+a^2+a+1)x^6+(a^3+1)x^4+x^3+(a^3+a^2+1)x+a^3+a^2+a$
- $y^2+(x^3+(a+1)x+a+1)y=(a^3+1)x^6+x^5+x^4+(a^2+1)x^3+(a^3+a+1)x^2+(a^3+1)x+1$
- $y^2+(x^3+a^3+a^2+1)y=(a^3+a+1)x^6+(a^3+a^2+1)x^4+x^3+(a^3+a^2+a)x+a^2+1$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a+1)x^6+x^5+x^4+ax^3+(a^2+1)x^2+(a^3+a^2+1)x+a^3+a^2+1$
- $y^2+(x^3+ax+a)y=(a^3+a^2)x^6+x^5+x^4+a^2x^3+(a^2+1)x^2+(a^3+a^2+a)x+a^3+1$
- $y^2+(x^3+a^3+a^2+a)y=(a^3+1)x^6+(a^3+a^2+a)x^4+x^3+(a^3+a+1)x+a$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a+1)x^6+(a^3+a^2)x^5+(a^3+a^2)x^4+(a^3+a^2+a+1)x^3+(a^2+1)x^2+a^2x+a^3$
- $y^2+(x^3+a^3+a^2+1)y=(a^3+a^2)x^6+(a^3+a^2+1)x^4+a^2x^3+(a^3+a^2+a)x+a^2+1$
- $y^2+(x^3+a^3+a^2+a)y=(a^3+a^2+1)x^6+(a^3+a^2+a)x^4+(a+1)x^3+(a^3+a+1)x+a^2+a+1$
- $y^2+(x^3+ax+a)y=(a^3+a^2)x^6+(a^3+a^2+a+1)x^5+(a^3+a^2+a+1)x^4+(a^3+a)x^3+(a^2+1)x^2+(a+1)x+a^3+a+1$
- $y^2+(x^3+(a+1)x+a+1)y=(a^3+a)x^6+(a^3+1)x^5+(a^3+1)x^4+(a^3+a)x^3+a^2x^2+(a^3+a^2+a)x+a$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a)x^3+a^2x+a^2$
- $y^2+(x^3+a^3+a+1)y=(a^3+a^2+1)x^6+(a^3+a+1)x^4+(a^2+1)x^3+(a^3+1)x+a^3+a^2+a$
- $y^2+(x^3+a^2x+a^2)y=(a^3+a^2+a)x^6+(a^3+a)x^5+(a^3+a)x^4+a^3x^3+x^2+(a^2+1)x+a^3+a$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a^2+a)x^6+(a^3+a^2+1)x^5+(a^3+a^2+1)x^4+a^3x^3+(a^3+a^2+a+1)x^2+(a^3+a+1)x+a^3$
- $y^2+(x^3+ax+a)y=a^3x^3+(a+1)x+a+1$
- $y^2+(x^3+a^2x+a^2)y=(a^3+a+1)x^5+(a^3+a+1)x^4+(a^2+a+1)x^3+(a^3+a+1)x+1$
- $y^2+(x^3+(a+1)x+a+1)y=(a^3+a+1)x^6+a^3x^5+a^3x^4+(a^3+a^2)x^3+x^2+ax+a^3$
- $y^2+(x^3+a^3+1)y=a^3x^6+(a^3+1)x^4+ax^3+(a^3+a^2+1)x+a+1$
- $y^2+(x^3+a^2x+a^2)y=(a^3+a^2+a+1)x^6+(a^3+a+1)x^5+(a^3+a+1)x^4+(a^3+a^2+a+1)x^3+ax^2+(a^3+a^2+1)x+a^2+1$
- $y^2+(x^3+(a+1)x+a+1)y=(a^3+a^2+a+1)x^3+ax+a$
- $y^2+(x^3+ax+a)y=(a^3+a^2+a)x^5+(a^3+a^2+a)x^4+(a^2+a)x^3+(a^3+a^2+a)x+1$
- $y^2+(x^3+a^2x+a^2)y=(a^3+a^2)x^3+(a^2+1)x+a^2+1$
- $y^2+(x^3+ax+a)y=(a^3+a+1)x^6+(a^3+a^2+a)x^5+(a^3+a^2+a)x^4+(a^3+a^2)x^3+(a^3+a)x^2+(a^3+1)x+a^3+a^2$
- $y^2+(x^3+(a^2+1)x+a^2+1)y=(a^3+a^2+1)x^5+(a^3+a^2+1)x^4+(a^2+a+1)x^3+(a^3+a^2+1)x+1$
- $y^2+(x^3+(a+1)x+a+1)y=(a^3+1)x^5+(a^3+1)x^4+(a^2+a)x^3+(a^3+1)x+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{12}}$.
Endomorphism algebra over $\F_{2^{4}}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{-3}, \sqrt{5})\). |
The base change of $A$ to $\F_{2^{12}}$ is 1.4096.h 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-15}) \)$)$ |
Base change
This isogeny class is not primitive. It is a base change from the following isogeny classes over subfields of $\F_{2^{4}}$.
Subfield | Primitive Model |
$\F_{2}$ | 2.2.ad_f |
$\F_{2}$ | 2.2.d_f |