Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 2 x + 3 x^{2} - 8 x^{3} + 16 x^{4}$ |
Frobenius angles: | $\pm0.168977707736$, $\pm0.618033150523$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.32832.1 |
Galois group: | $D_{4}$ |
Jacobians: | $4$ |
Isomorphism classes: | 4 |
This isogeny class is simple and 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
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $10$ | $300$ | $3310$ | $69600$ | $1175050$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $3$ | $19$ | $51$ | $271$ | $1143$ | $4147$ | $16467$ | $66271$ | $261879$ | $1045699$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 4 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+(x^2+x+a)y=ax^5+x^3+(a+1)x+1$
- $y^2+(x^2+x+a+1)y=(a+1)x^5+x^3+ax+1$
- $y^2+(x^2+x+a)y=ax^5+ax^4+ax^2+a$
- $y^2+(x^2+x+a+1)y=(a+1)x^5+ax^4+ax^2+1$
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{2}}$.
Endomorphism algebra over $\F_{2^{2}}$The endomorphism algebra of this simple isogeny class is 4.0.32832.1. |
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.c_d | $2$ | 2.16.c_j |