Invariants
Base field: | $\F_{2^{2}}$ |
Dimension: | $2$ |
L-polynomial: | $1 - 3 x + 6 x^{2} - 12 x^{3} + 16 x^{4}$ |
Frobenius angles: | $\pm0.150432950460$, $\pm0.544835058382$ |
Angle rank: | $2$ (numerical) |
Number field: | 4.0.3757.1 |
Galois group: | $D_{4}$ |
Jacobians: | $2$ |
This isogeny class is simple and geometrically simple, primitive, not ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.
Newton polygon
$p$-rank: | $1$ |
Slopes: | $[0, 1/2, 1/2, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $8$ | $304$ | $3656$ | $61408$ | $1141688$ |
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $2$ | $20$ | $56$ | $240$ | $1112$ | $4304$ | $16424$ | $65728$ | $263576$ | $1048880$ |
Jacobians and polarizations
This isogeny class contains the Jacobians of 2 curves (of which all are hyperelliptic), and hence is principally polarizable:
- $y^2+xy=ax^5+x^3+x$
- $y^2+xy=(a+1)x^5+x^3+x$
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 4.0.3757.1. |
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.d_g | $2$ | 2.16.d_ae |