Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 + 3 x^{2} + 16 x^{4}$ |
Frobenius angles: | $\pm0.311178646770$, $\pm0.688821353230$ |
Angle rank: | $1$ (numerical) |
Number field: | \(\Q(\sqrt{5}, \sqrt{-11})\) |
Galois group: | $C_2^2$ |
Jacobians: | $6$ |
Isomorphism classes: | 12 |
This isogeny class is simple but not geometrically 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $20$ | $400$ | $3980$ | $78400$ | $1050500$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $5$ | $23$ | $65$ | $303$ | $1025$ | $3863$ | $16385$ | $65503$ | $262145$ | $1052423$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 6 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x+a)y=x^5+a$
- $y^2+(x^2+x+a)y=x^5+(a+1)x^4+(a+1)x^2$
- $y^2+(x^2+x)y=(a+1)x^5+(a+1)x^3+ax^2+ax$
- $y^2+(x^2+x+a+1)y=x^5+a+1$
- $y^2+(x^2+x+a+1)y=x^5+(a+1)x^4+(a+1)x^2+a$
- $y^2+(x^2+x)y=ax^5+ax^3+(a+1)x^2+(a+1)x$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{4}}$.
Endomorphism algebra over $\F_{2^{2}}$The endomorphism algebra of this simple isogeny class is \(\Q(\sqrt{5}, \sqrt{-11})\). |
The base change of $A$ to $\F_{2^{4}}$ is 1.16.d 2 and its endomorphism algebra is $\mathrm{M}_{2}($\(\Q(\sqrt{-55}) \)$)$ |
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 |
---|---|---|
2.4.a_ad | $4$ | 2.256.bu_bob |